Stefan Rolewicz Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Metric Linear Spaces
D. Reidel P...

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Stefan Rolewicz Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

Metric Linear Spaces

D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP

Dordrecht/Boston/Lancaster

PWN-Polish Scientific Publishers Warszawa

Library of Congress Cataloging in Publication Data Rolewicz, Stefan. Metric linear spaces.

(Mathematics and its applications. East European series; v. ) Bibliography: p. Includes index. 1. Metric spaces. 2. Locally convex spaces. I. Title. 11. Series : Mathematics and its applications (D. Reidel Publishing Company). East European series; v. QA611.28.R6513 1984 ISBN 90-277-1480-0 (Reidel)

514'.3

83-24541

First edition published in Monografie Matematyczne series by Paristwowe Wydawnictwo Naukowe, Warszawa 1972 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland and PWN -- Polish Scientific Publishers, Miodowa 10, 00-251 Warszawa, Poland

Distributors for Albania, Bulgaria, Cuba, Czechoslovakia, German Democratic Republic, Hungary, Korean People's Democratic Republic, Mongolia, People's Republic of China, Poland, Romania, the U.S.S.R., Vietnam, and Yugoslavia ARS POLONA Krakowskie Przedmiescie 7, 00-068 Warszawa 1, Poland Distributors for the U.S.A. and Canada Kluwer Academic Publishers, 190 Old Derby Street, Hingham, MA 02043, U.S.A. Distributors for all other countries Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland

All Rights Reserved

© 1985 by PWN - Polish Scientific Publishers - Warszawa. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in Poland by, D.S.P.

Editor's Preface

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years : measure theory is used (non-trivially) in regional and theoretical economics ; algebraic geometry interacts with physics ; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering ; and prediction and electrical engineering can use Stein spaces. This series of books, Mathematics and Its Applications, is devoted to such (new) interrelations as exempli gratia :

- a central concept which plays an important role in several different mathematical and/or scientific specialized areas ;

- new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields. V

VI

Editor's Preface

Because of the wealth of scholary research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA programme.

The present volume in the MIA Eastern Europe series is devoted to non-locally convex spaces. It is a thoroughly revised, augmented and corrected version of the first edition of 1972 in Monografie Matematyczne

(Mathematical Monographs). At that time already, non-locally convex spaces had become very important, e.g. in connection with integral operators and stochastic processes. In the 10 years since then several new applications have appeared (also to other fields) and a great many new results have been obtained, justifying this new augmented edition.

Krimpena/dlJssel, July, 1982.

MICHIEL HAZEWINKEL

Table of Contents

Editor's Preface

V

Preface

X

CHAPTER 1. Basic Facts on Metric Linear Spaces 1.1. Definition of Metric Linear Spaces and the Theorem on the Invariant Norm 1.2. Modular Spaces 1.3. Examples of Metric Linear Spaces 1.4. Complete Metric Linear Spaces 1.5. Complete Metric Linear Spaces. Examples 1.6. Separable Spaces 1.7. Topological Linear Spaces

CHAPTER 2. Linear Operators 2.1. Basic Properties of Linear Operators 2.2. Banach-Steinhaus Theorem for F-Spaces 2.3. Continuity of the Inverse Operator in F-Spaces 2.4. Linear Dimension and the Existence of a Universal Space

I 1

6 10 18

23 26 33

36 36 39 42 44

2.5. Linear Codimension and the Existence of a Co-Universal Space 2.6. Bases in F-Spaces 2.7. Solid Metric Linear Spaces and General Integral Operators

62

CHAPTER 3. Locally Pseudoconvex and Locally Bounded Spaces 3.1. Locally Pseudoconvex Spaces 3.2. Locally Bounded Spaces

89 89 95

VII

67 77

VIII

Table of Contents

3.3. Bounded Sets in Spaces N(L(Q, E, µ)) 3.4. Calculation of the Modulus of Concavity

107

of Spaces

N(L(Q, E, µ)) 3.5. Integrations of Functions with Values in F-Spaces 3.6. Vector-Valued Measures 3.7. Integration with Respect to an Independent Random Measure 3.8. Unconditional Convergence of Series 3.9. Invariant A(X) 3.10. C-Sequences and C-Spaces 3.11. Locally Bounded Algebras 3.12. Law of Large Numbers in Locally Bounded Spaces

111

120 127 145 152 157 165 173 183

CHAPTER 4. Existence and Non-existence of Continuous Linear Functionals and Continuous Linear Operators 187 4.1. Continuous Linear Functionals and Open Convex Sets 187 4.2. Existence and Non-Existence of Continuous Linear Functionals

193

4.3. General Form of Continuous Linear Functionals in Concrete Banach Spaces 199 4.4. Continuous Linear Functionals in Bo-Spaces 202 4.5. Non-Existence of Non-Trivial Compact Operators 206 4.6. Existence of Rigid Spaces 210 CHAPTER 5. Weak Topologies 5.1. Convex Sets and Locally Convex Topological Spaces 5.2. Weak Topologies. Basic Properties 5.3. Weak Convergence 5.4. Example of an Infinite-Dimensional Banach Space which is not Isomorphic to Its Square 5.5. Extreme Points 5.6. Existence of a Convex Compact Set without Extreme Poinst

221 221

CHAPTER 6. Montel and Schwartz Spaces 6.1. Compact Sets in F-Spaces 6.2. Montel Spaces 6.3. Schwartz Spaces

249 249

226 230 234 238 241

251

255

Table of Contents

IX

6.4. Characterization of Schwartz Spaces by a Property of FNorms 6.5. Approximative Dimension 6.6. Diametral Dimension 6.7. Isomorphism and Near-Isomorphism of the Cartesian Products

259 263 274 288

CHAPTER 7. Nuclear Spaces. Theory 7.1. Definition and Basic Properties of Nuclear Spaces 7.2. Nuclear Operators and Nuclear Locally Convex Spaces 7.3. Unconditional and Absolute Convergence 7.4. Bases in Nuclear Spaces 7.5. Spaces with Regular Bases 7.6. Universal Space for Nuclear Spaces

296 296 308 314 326

CHAPTER 8. Nuclear Spaces. Examples and Applications 8.1. Spaces of Infinitely Differentiable Functions 8.2. Spaces of Holomorphic Functions 8.3. Spaces of Holomorphic Functions. Continuation 8.4. Spaces of Dirichlet Series 8.5. Cauchy-Hadamard Formula for Kothe Power Spaces

344 344 354

333

340

361 371

376

CHAPTER 9. F-Norms and Isometries in F-Spaces 9.1. Properties. of F-Norms 9.2. Spaces,with Bounded Norms 9.3. Isometries and Rotations 9.4. Isometrical Embeddings in Banach Spaces 9.5. Group of Isometries in Finite-Dimensional Spaces 9.6. Spaces with Transitive and Almost Transitive Norms 9.7. Convex Transitive Norms 9.8. The Maximality of Symmetric Norms 9.9. Universality with Respect to Isometry

385 385 389 390 400 408 409 415

References Subject Index

434 449 456 459

Author Index List of Symbols

421

426

Preface

The definition of linear metric spaces was given by Frechet (1926). The basic facts in the theory of linear metric spaces were proved before 1940 (largely by Banach and his collaborators). At the beginning the investigations concentrated mainly on the theory of normed spaces, and the appearance of the theory of distribution in-

duced fast progress of investigations in the theory of locally convex spaces.

The development of the theory of integral operators and the theory of stochastic processes has aroused interest in the theory of non-locally convex spaces. Several papers dealing with this topic have been published, but this book has been the first monograph devoted to the subject. The first edition of this book was published ten years ago in Monografie Matematyczne, due to the encouragement of Professors K. Borsuk and K. Kuratowski. Since then the theory of non-locally convex spaces has been intensively developed. New applications of the theory have been discovered in probability theory, in the theory of integral operators and in analytic functions. Several of the open problems described in the first edition have been solved. For these reasons the second edition is a rewritten and enlarged version of the original book. The main changes are as follows : In Sections 2.3 and 2.4, Kalton's results about the existence of universal and co-universal F-spaces are presented. Section 2.7 contains a brief description of the theory of solid metric linear spaces and the theory of general integral operators (the results of Aronszajn, Szeptycki, Luxemburg and others). In Sections 3.6 and 3.7 the theory of integration with respect to vectorvalued measures with values in F-spaces is discussed (the results of Drew-

x

Preface

XI

nowski, Labuda, Maurey, Pisier, Ryll-Nardzewski, Urbanik, Woyczyiiski and others). The topic is closely connected with the theory of stochastic processes. Section 3.11 gives a concise description of the theory of locally bounded algebras developed by Zelazko. On the basis of this theory an extension of the Wiener theorem is presented. Section 3.12 contains the results of Sundaresan and Woyczynski concerning the convergence of series of independent random variables in locally bounded spaces. Section 4.5 contains Pallaschke's result, showing that in certain Orlicz spaces there are no linear compact operators different from 0. In Section 4.6 we present an example of a rigid space (i.e., such that each continuous linear operator is of the form aI, where I is an identity) constructed by Roberts and modified by Kalton and Roberts.

Section 5.6 contains an example of a compact set without extreme points constructed by Roberts. In Section 6.6 we present the results of Turpin showing that in Orlicz spaces every bounded set with approximative diameters tending to 0 is precompact. Section 6.7 contains extensions of Zahariuta results concerning an isomorphism and a near-isomorphism of Cartesian products of spaces to the case of locally p-convex spaces. Section 7.1-contains the theorem of Ligaud, stating that every nuclear locally pseudoconvex space is locally convex. There is also an example of a nuclear space with a trivial dual, constructed by Ligaud. Regular bases are considered in Section 7.5. The section contains the theorem of Crone and Robinson, stating that in nuclear Bo spaces with regular bases all bases are quasi-equivalent. We give a new proof of this theorem, based on a lemma of Kondakov. The section also contains the results of Djakov and Dragilev. Section 9.3 contains the result of Mankiewicz, stating that in strictly galbed spaces with a strong Krein-Milman property all isometries are affine.

Section 9.7 contains the results concerning the maximality of the standard norm in the space of continuous functions on locally compact spaces (the results of Cowie, Wood, Kalton and Wood).

XII

Preface

The reader is expected to be familiar with elementary facts in topology and in linear algebra. The knowledge of functional analysis is not required. For this reason the book contains facts about Banach spaces, useful in further considerations. During the preparation of the second edition I was helped by several of my colleagues, who offered me their advice and criticism. Here is a par-

tial list of those to whom I owe heartfelt thanks : S. Dierolf, T. Dobrowolski, V. Eberhardt, W. Herer, N.J. Kalton, J. J. Koliha, Z. Lipecki, W. Lipski, Ph. Turpin, W. Woyczyfiski. I also wish to express my gratitude to C. Bessaga for his careful and penetrating perusal of the manuscript and his valuable remarks, and to V.P. Kondakov and G.V. Wood, who gave me access to their still unpublished results and consented to their being included in this book. Warszawa, April 1982

S T E F A N R O L E W IC Z

Chapter 1

Basic Facts on Metric Linear Spaces

I.I. DEFINITION OF METRIC LINEAR SPACES AND THE THEOREM ON THE INVARIANT NORM

Let X be a linear space over either complex or real numbers. The main part of our considerations will be the same in both cases; therefore, by the

term linear space we shall understand simultaneously the real and the complex case. When needed, we shall specify that either a complex linear space or a real linear space is considered. The operation of addition of elements x, y will be denoted, as usual, by x+y. The operation of multiplication of an element x by a scalar t will be denoted by tx. By A+B we shall denote the set {x+y: x e A, y e B}. By

to we shall denote the set {tx: x e A}. Suppose that in the space X we are given a two-argument non-negative real-valued function p(x, y) satisfying the following conditions :

(ml) p(x,y) =0 if and only if x = y, (m2) p(x, y) = p(y, x), (m3) p(x, y)

Metric Linear Spaces

D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP

Dordrecht/Boston/Lancaster

PWN-Polish Scientific Publishers Warszawa

Library of Congress Cataloging in Publication Data Rolewicz, Stefan. Metric linear spaces.

(Mathematics and its applications. East European series; v. ) Bibliography: p. Includes index. 1. Metric spaces. 2. Locally convex spaces. I. Title. 11. Series : Mathematics and its applications (D. Reidel Publishing Company). East European series; v. QA611.28.R6513 1984 ISBN 90-277-1480-0 (Reidel)

514'.3

83-24541

First edition published in Monografie Matematyczne series by Paristwowe Wydawnictwo Naukowe, Warszawa 1972 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland and PWN -- Polish Scientific Publishers, Miodowa 10, 00-251 Warszawa, Poland

Distributors for Albania, Bulgaria, Cuba, Czechoslovakia, German Democratic Republic, Hungary, Korean People's Democratic Republic, Mongolia, People's Republic of China, Poland, Romania, the U.S.S.R., Vietnam, and Yugoslavia ARS POLONA Krakowskie Przedmiescie 7, 00-068 Warszawa 1, Poland Distributors for the U.S.A. and Canada Kluwer Academic Publishers, 190 Old Derby Street, Hingham, MA 02043, U.S.A. Distributors for all other countries Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland

All Rights Reserved

© 1985 by PWN - Polish Scientific Publishers - Warszawa. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in Poland by, D.S.P.

Editor's Preface

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years : measure theory is used (non-trivially) in regional and theoretical economics ; algebraic geometry interacts with physics ; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering ; and prediction and electrical engineering can use Stein spaces. This series of books, Mathematics and Its Applications, is devoted to such (new) interrelations as exempli gratia :

- a central concept which plays an important role in several different mathematical and/or scientific specialized areas ;

- new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields. V

VI

Editor's Preface

Because of the wealth of scholary research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA programme.

The present volume in the MIA Eastern Europe series is devoted to non-locally convex spaces. It is a thoroughly revised, augmented and corrected version of the first edition of 1972 in Monografie Matematyczne

(Mathematical Monographs). At that time already, non-locally convex spaces had become very important, e.g. in connection with integral operators and stochastic processes. In the 10 years since then several new applications have appeared (also to other fields) and a great many new results have been obtained, justifying this new augmented edition.

Krimpena/dlJssel, July, 1982.

MICHIEL HAZEWINKEL

Table of Contents

Editor's Preface

V

Preface

X

CHAPTER 1. Basic Facts on Metric Linear Spaces 1.1. Definition of Metric Linear Spaces and the Theorem on the Invariant Norm 1.2. Modular Spaces 1.3. Examples of Metric Linear Spaces 1.4. Complete Metric Linear Spaces 1.5. Complete Metric Linear Spaces. Examples 1.6. Separable Spaces 1.7. Topological Linear Spaces

CHAPTER 2. Linear Operators 2.1. Basic Properties of Linear Operators 2.2. Banach-Steinhaus Theorem for F-Spaces 2.3. Continuity of the Inverse Operator in F-Spaces 2.4. Linear Dimension and the Existence of a Universal Space

I 1

6 10 18

23 26 33

36 36 39 42 44

2.5. Linear Codimension and the Existence of a Co-Universal Space 2.6. Bases in F-Spaces 2.7. Solid Metric Linear Spaces and General Integral Operators

62

CHAPTER 3. Locally Pseudoconvex and Locally Bounded Spaces 3.1. Locally Pseudoconvex Spaces 3.2. Locally Bounded Spaces

89 89 95

VII

67 77

VIII

Table of Contents

3.3. Bounded Sets in Spaces N(L(Q, E, µ)) 3.4. Calculation of the Modulus of Concavity

107

of Spaces

N(L(Q, E, µ)) 3.5. Integrations of Functions with Values in F-Spaces 3.6. Vector-Valued Measures 3.7. Integration with Respect to an Independent Random Measure 3.8. Unconditional Convergence of Series 3.9. Invariant A(X) 3.10. C-Sequences and C-Spaces 3.11. Locally Bounded Algebras 3.12. Law of Large Numbers in Locally Bounded Spaces

111

120 127 145 152 157 165 173 183

CHAPTER 4. Existence and Non-existence of Continuous Linear Functionals and Continuous Linear Operators 187 4.1. Continuous Linear Functionals and Open Convex Sets 187 4.2. Existence and Non-Existence of Continuous Linear Functionals

193

4.3. General Form of Continuous Linear Functionals in Concrete Banach Spaces 199 4.4. Continuous Linear Functionals in Bo-Spaces 202 4.5. Non-Existence of Non-Trivial Compact Operators 206 4.6. Existence of Rigid Spaces 210 CHAPTER 5. Weak Topologies 5.1. Convex Sets and Locally Convex Topological Spaces 5.2. Weak Topologies. Basic Properties 5.3. Weak Convergence 5.4. Example of an Infinite-Dimensional Banach Space which is not Isomorphic to Its Square 5.5. Extreme Points 5.6. Existence of a Convex Compact Set without Extreme Poinst

221 221

CHAPTER 6. Montel and Schwartz Spaces 6.1. Compact Sets in F-Spaces 6.2. Montel Spaces 6.3. Schwartz Spaces

249 249

226 230 234 238 241

251

255

Table of Contents

IX

6.4. Characterization of Schwartz Spaces by a Property of FNorms 6.5. Approximative Dimension 6.6. Diametral Dimension 6.7. Isomorphism and Near-Isomorphism of the Cartesian Products

259 263 274 288

CHAPTER 7. Nuclear Spaces. Theory 7.1. Definition and Basic Properties of Nuclear Spaces 7.2. Nuclear Operators and Nuclear Locally Convex Spaces 7.3. Unconditional and Absolute Convergence 7.4. Bases in Nuclear Spaces 7.5. Spaces with Regular Bases 7.6. Universal Space for Nuclear Spaces

296 296 308 314 326

CHAPTER 8. Nuclear Spaces. Examples and Applications 8.1. Spaces of Infinitely Differentiable Functions 8.2. Spaces of Holomorphic Functions 8.3. Spaces of Holomorphic Functions. Continuation 8.4. Spaces of Dirichlet Series 8.5. Cauchy-Hadamard Formula for Kothe Power Spaces

344 344 354

333

340

361 371

376

CHAPTER 9. F-Norms and Isometries in F-Spaces 9.1. Properties. of F-Norms 9.2. Spaces,with Bounded Norms 9.3. Isometries and Rotations 9.4. Isometrical Embeddings in Banach Spaces 9.5. Group of Isometries in Finite-Dimensional Spaces 9.6. Spaces with Transitive and Almost Transitive Norms 9.7. Convex Transitive Norms 9.8. The Maximality of Symmetric Norms 9.9. Universality with Respect to Isometry

385 385 389 390 400 408 409 415

References Subject Index

434 449 456 459

Author Index List of Symbols

421

426

Preface

The definition of linear metric spaces was given by Frechet (1926). The basic facts in the theory of linear metric spaces were proved before 1940 (largely by Banach and his collaborators). At the beginning the investigations concentrated mainly on the theory of normed spaces, and the appearance of the theory of distribution in-

duced fast progress of investigations in the theory of locally convex spaces.

The development of the theory of integral operators and the theory of stochastic processes has aroused interest in the theory of non-locally convex spaces. Several papers dealing with this topic have been published, but this book has been the first monograph devoted to the subject. The first edition of this book was published ten years ago in Monografie Matematyczne, due to the encouragement of Professors K. Borsuk and K. Kuratowski. Since then the theory of non-locally convex spaces has been intensively developed. New applications of the theory have been discovered in probability theory, in the theory of integral operators and in analytic functions. Several of the open problems described in the first edition have been solved. For these reasons the second edition is a rewritten and enlarged version of the original book. The main changes are as follows : In Sections 2.3 and 2.4, Kalton's results about the existence of universal and co-universal F-spaces are presented. Section 2.7 contains a brief description of the theory of solid metric linear spaces and the theory of general integral operators (the results of Aronszajn, Szeptycki, Luxemburg and others). In Sections 3.6 and 3.7 the theory of integration with respect to vectorvalued measures with values in F-spaces is discussed (the results of Drew-

x

Preface

XI

nowski, Labuda, Maurey, Pisier, Ryll-Nardzewski, Urbanik, Woyczyiiski and others). The topic is closely connected with the theory of stochastic processes. Section 3.11 gives a concise description of the theory of locally bounded algebras developed by Zelazko. On the basis of this theory an extension of the Wiener theorem is presented. Section 3.12 contains the results of Sundaresan and Woyczynski concerning the convergence of series of independent random variables in locally bounded spaces. Section 4.5 contains Pallaschke's result, showing that in certain Orlicz spaces there are no linear compact operators different from 0. In Section 4.6 we present an example of a rigid space (i.e., such that each continuous linear operator is of the form aI, where I is an identity) constructed by Roberts and modified by Kalton and Roberts.

Section 5.6 contains an example of a compact set without extreme points constructed by Roberts. In Section 6.6 we present the results of Turpin showing that in Orlicz spaces every bounded set with approximative diameters tending to 0 is precompact. Section 6.7 contains extensions of Zahariuta results concerning an isomorphism and a near-isomorphism of Cartesian products of spaces to the case of locally p-convex spaces. Section 7.1-contains the theorem of Ligaud, stating that every nuclear locally pseudoconvex space is locally convex. There is also an example of a nuclear space with a trivial dual, constructed by Ligaud. Regular bases are considered in Section 7.5. The section contains the theorem of Crone and Robinson, stating that in nuclear Bo spaces with regular bases all bases are quasi-equivalent. We give a new proof of this theorem, based on a lemma of Kondakov. The section also contains the results of Djakov and Dragilev. Section 9.3 contains the result of Mankiewicz, stating that in strictly galbed spaces with a strong Krein-Milman property all isometries are affine.

Section 9.7 contains the results concerning the maximality of the standard norm in the space of continuous functions on locally compact spaces (the results of Cowie, Wood, Kalton and Wood).

XII

Preface

The reader is expected to be familiar with elementary facts in topology and in linear algebra. The knowledge of functional analysis is not required. For this reason the book contains facts about Banach spaces, useful in further considerations. During the preparation of the second edition I was helped by several of my colleagues, who offered me their advice and criticism. Here is a par-

tial list of those to whom I owe heartfelt thanks : S. Dierolf, T. Dobrowolski, V. Eberhardt, W. Herer, N.J. Kalton, J. J. Koliha, Z. Lipecki, W. Lipski, Ph. Turpin, W. Woyczyfiski. I also wish to express my gratitude to C. Bessaga for his careful and penetrating perusal of the manuscript and his valuable remarks, and to V.P. Kondakov and G.V. Wood, who gave me access to their still unpublished results and consented to their being included in this book. Warszawa, April 1982

S T E F A N R O L E W IC Z

Chapter 1

Basic Facts on Metric Linear Spaces

I.I. DEFINITION OF METRIC LINEAR SPACES AND THE THEOREM ON THE INVARIANT NORM

Let X be a linear space over either complex or real numbers. The main part of our considerations will be the same in both cases; therefore, by the

term linear space we shall understand simultaneously the real and the complex case. When needed, we shall specify that either a complex linear space or a real linear space is considered. The operation of addition of elements x, y will be denoted, as usual, by x+y. The operation of multiplication of an element x by a scalar t will be denoted by tx. By A+B we shall denote the set {x+y: x e A, y e B}. By

to we shall denote the set {tx: x e A}. Suppose that in the space X we are given a two-argument non-negative real-valued function p(x, y) satisfying the following conditions :

(ml) p(x,y) =0 if and only if x = y, (m2) p(x, y) = p(y, x), (m3) p(x, y)

We say that a set U C X is balanced if, for any number a, ja! < 1, a U e U. Any neighbourhood of zero W contains a balanced neighbourhood of zero U. Indeed, the continuity of multiplication by numbers implies that there are a neighbourhood of zero V and a positive number E such that aV e V provided Ia! < s. Therefore the set I

Chapter 1

2

U=UaV lol < E

is contained in W. It is easy to verify that U is a balanced neighbourhood of zero. We say that two metrics, p(x, y) and p'(x, y), are equivalent if the topologies induced by those metrics are equivalent, in other words : if for any

r > 0 and x e X there are 6, b' > 0 such that Y: P'(x, Y) < 8 C {y: p(x, Y) < r}, Y: P(x, Y) < a') C {Y: P'(x, Y) < q. A sequence {xn} of elements of .x is said to tend to an element x e X (or to be convergent to x) with respect to the metric p(x, y) if

limp(xn, x) = 0. n-*oo

We shall write this as Xn-*x. P

If no misunderstanding can arise, we shall say briefly that xn tends to x (xn is convergent to x) and write xn-x. A metric pl is said to be stronger than a metric p, if x-*y implies x-y. P,

P

If pl is stronger than p and simultaneously p is stronger than pl, we say that the metrics p and pl are equivalent. A metric p(x, y) is called invariant if

p(x+z, y+z) = p(x, y) for all x, y, z e X. THEOREM 1.1.1 (Kakutani, 1936). Let(X, p) be a metric linear space. Then there is an invariant metric p'(x, y) equivalent to the original metric p(x, y).

Proof. Let U be an arbitrary balanced neighbourhood of zero. Write

U(1) = U

and

U(n) = U+...+U. n-fold

The continuity of addition implies that there is a neighbourhood of

zero U(2) such that U(;)+U(2) C U(1). Of course, without loss of generality we can assume that U(z) is balanced and that U(12) C K112= {x: p(x, 0) <

a}.

Basic Facts on Metric Linear Spaces

3

Arguing in the same manner, we can find by induction balanced neighbourhoods of zero U(2) such that (1.1.1) U(-n)CK,

jx: p(x,0)<

2n}.

(1.1.2)

Let r be an arbitrary dyadic number

r = n +2

z

+ 22 +... + 2n,

where ai is equal either to 0 or to 1, i = 1, 2, ..., n. Let

U(r) = U(n)+a,U(2)+...+a-U(Zn). Obviously U(r) is a balanced neighbourhood of zero. By (1.1.1), for two arbitrary dyadic numbers r1, r2 we have U (r1+r2) D U(rl)+ U(r2)

(1.1.3)

Let

p'(x, y) = inf (r: x- y e U(r)j. We shall show that p'(x, y) is an invariant metric equivalent to the metric

p(x,y). Indeed,

p'(x+z, y+z) = inf{r: (x+z)-(y+z) e U(r)} = inf{r: x-y e U(r)} = p'(x, y). Since U(r) are balanced, we have P'(x, Y) = P'(y, X) Inclusion (1.1.3) implies the triangle inequality (m3). Indeed,

p'(x, z)--p'(z, y) = inf{r1: x-z e U(rl)}+inf{r2: z-y e U(r2)1 = inf{r1+r2: x-z e U(rl), z-y e U(r2)} inf r1+r2: (x-z)+(z-y) e U(rl)+ U(r2)} inf r1+r2: x-y e U(rl+r2)} = inf{r: x-y e U(r)} = p'(x, y). We have proved that p'(x, y) is invariant and that it satisfies properties (m2) and (m3). We shall now show that

Jim p' (x., x) = 0 if and only if lim p(xn, x) = 0.

Chapter 1

4

This would imply thatp'(x,y) satisifies (ml). Thus p'(x,y) is a metric equiv-

alent to the metric p(x,y). Let p'(xk, x) tend to 0. Then (1.1.2) implies that p(xk-x, 0) converges to 0. The continuity of addition implies lim p(xk,x) = 0. k-.,o

On the other hand, if p (xk, x) tends to 0, then the continuity of addition

implies that xk-x converges to 0. Since U (--) are neighbourhoods of

zero, for an arbitrary n there is an index ko such that for k > ko,

xk-xe U 2n 1. 1

This implies that for k > ko p'(xk, x) < 2n . From the arbitrariness of n it follows that lim p'(xk, x) = 0. k-.oo

Let X be a linear space. A non-negative valued function IIxII defined on X is called an F-norm (or briefly a norm) if it satisfies the following conditions :

(nl) Ilxll = 0 if and only if x = 0, (n2) Ilaxll = lIxll for all a, lal = 1, (n3) llx+yli < IIxII+IIYIl, (n4) Ha n xl I -> 0 provided an -* 0,

(n5) llaxnl I ->0 provided xn -*0, 0 provided an -* 0, xn -*0, (n6) Ilan xnll PROPOSITION I.I.I. If llxn-xll -* 0 and an --> a, then Ilanxn-axll ->-0. Proof.

Ilanxn-axll < II(an-a) (x.-x) II+II(an-a)xll+Ila(xn-x)Il. Since the right-hand terms tend to 0 by (n4)-(n6), the left-hand term also tends to 0. Each F-norm IIxII induces an invariant metric p(x,y) by following formula

AX, Y) = lIx-YII

(1.1.4)

On the other hand, if Xis a metric linear space over reals, then p(x, 0) = IIxII is an F-norm, provided that p is invariant.

Basic Facts on Metric Linear Spaces

5

A linear space equipped with an F-norm II II is called an F*-space (see Banach, 1932). An F*-space equipped with an F-norm IIzII we shall denote II) or briefly X. Let two norms II II and I II1 be defined on the same space X. The norm II1 is said to be stronger than the norm II II (equivalent to the norm II II)

(X, 11

II

if the corresponding invariant metric p1 is stronger than (equivalent to) the corresponding invariant metric p. Let X be an F*-space and let Y be a linear subset of X. It is obvious that Y is also an F*-space with the F-norm obtained by the restriction of the original F-norm in X to Y. Closed linear subsets are called subspaces. Let (X, II 11) be an F*-space and let Y be a subspace of the space X. By X/Y we denote the quotient space, i.e. the space which has cosets with respect to Y as elements. We define the norm of the coset Z in the following way : IIZII = inf{IIzII: z c- Z).

It is easy to verify that IIZII is an F-norm. In fact IIZII = 0 if and only if Z = Y, i.e. Z is the zero element of the quotient space. Let Ial = 1. Then IIaZII = inf{Ilazll : z e Z} = inf{IIzII : z c Z} = IIZII.

Let Z1, Z2 be two arbitrary cosets. Then IIZI+2211 = inf{IIZ1+z2II : Z1 E Z1, Z2 E Z2} < inf IIZ1II+IIz2II : ZL E Z1, Z2 E Z2} = inf{Ilzlll: z1 a Z1}+inf{IIz2II: z2 E Z2}

= IIZII+IIZ2II

If an -> 0, then for any fixed z e Z llanZIj = inf{Ilanzll: z e Z} < Ilanzoll --±0. If llZnll ->0, then there is a sequence zn E Zn, IIznII -> 0. Therefore for each bounded sequence an Ilanznll

which implies (n5) and (n6). The quotient space X/Y with the F-norm IIZII is called a quotient F*space.

Chapter 1

6

Let a system of n F*-spaces (Xi, 11 I Is), i = 1, ... , n, be given. By theprod-

uct of those spaces we mean the space of all systems x = {x1i ..., xn}, or xs a X{, i = 1, ..., n with the norm n

Ilxll =

s=

Ilx{IIi

The product space will be denoted by (X1 x briefly X1 x

... x X,, 1111,+ ... +11 11n)

... x Xn.

1.2. MODULAR SPACES

Let X be a linear space. A modular is a non-negative valued function p(x) defined on X and admitting also value +oo satisfying the following conditions :

(mdl)p(x)=0ifandonly ifx=0, (md2) p(ax) = p(x) provided lal = 1, (md3) p(ax+by) < p(x)+p(y) provided a,b > 0, a+b = 1, (md4) p(anx)--0 if a,, ->0 and p(x) < +oo, (see Nakano, 1950; Musielak, Orlicz, 1959, 1959b; Musielak, 1978). It moreover (md5) p(axn)->0 provided p(xn)->.0, the modular p (x) is called metrizing. In this book only metrizing modulars will be considered. Thus, for brevity, we shall call them simply modulars Putting y = 0 in (md3) we obtain

p(ax) < p(x)

if 0 < a < 1,

(1.2.1)

i.e. p (ax) is a non-decreasing function of the argument a for non-negative

a and all x e X. Formula (1.2.1) implies that (md5) is equivalent to the following (md5') p(xn)->0 if and only if p(2xn)-*0. Let X. be the set of all x e X such that there exists a positive number k

such that p (kx) < +oo. PROPOSITION 1.2.1. The set XP is linear.

Proof. Let x e X, and let t be a scalar. From the definition of X. it follows that there is a positive number k such that

Basic Facts on Metric Linear Spaces

P(kx)=p(

7

k

tx

Itl x)

Therefore tx a XP. Let x, y e XP. The definition of XP implies that there are such positive numbers kx and kk that

p(kyy) <+oo.

P(kxx) <+oo,

Let k = min (kx, ky). Then by (md3) we obtain P

(k

(x+y))

Hence x+y e XP. An F-norm IIxII need not be a modular, but if

if 0 < a < 1,

Ilaxll < IIxII

(1.2.2)

then the norm IIxII is a modular. A norm satisfying (1.2.2) is called non-decreasing.

The following two theorems show the relation between F-norms and modulars. THEOREM 1.2.2. Let (X, II ID be an F-space. Then the norm 114,

sup 11tX11

o

1

is equivalent to the original norm IIxII, and it is non-decreasing (i.e. it is a modular). Proof. If x = 0, then IIxII' = sup IItxII = 0. If IIxII' = 0, then IIxII = 0 (since 05a<_1

IIxII < IIxII') and x = 0. If Jai = 1

IIax!I = sup Iltaxll = sup IItxII = IIxII o<_e_*
0_
Let x, y e X. Then
IIx+yii' = sup IIt(x+y) II < sup IItxII+sup IItYIl *

+e+rl n/

Ce+r!
Therefore Ilx+yll < E+?I < IIxII+IIYII+2S.
The arbitrariness of 6 implies the triangle inequality (n3). We shall now show that llxnl H->0 if and only if p(xn)->0.
Suppose that p(xn)->0. Let e be an arbitrary positive number. By(md5)
p (-) -0. Therefore, there is an N such that for n > N, p (--) <e. e e
Ilxnll->0. Hence llxnll < e. The arbitrariness of e implies that On the other hand, let Ilxnll < 1, and let a be an arbitrary number such
that llx.ll < a < 1. Then
in LP(am,n) for p > 1. Example 1.3.10 Let X be a linear space. Let (x, y) be a two-argument scalar-valued function satisfying the following conditions : 0, (i l) (x, x)
(i2) (x, x) = 0 if and only if x = 0, (i3) (x1+x2,Y) = (x1,Y)+(x2,Y),
(i4) (ax,y) = a(x,y) for all scalars a, (i5) (x, y) = (y, x), where a denotes the conjugate number to the number a.
Chapter 1
18
The function (x, y) is called an inner product. The space X with the inner product is called a pre-Hilbert space. It is easy to verify that Ilxll =1/(x, x) is an F-norm.
1.4. COMPLETE METRIC LINEAR SPACES
Let X be a metric space with a metric p (x, y). A sequence {xn} of elements
of X is said to be a Cauchy sequence or to satisfy the Cauchy condition, or to be fundamental if lim p(xn, xm) = 0, n,m--+00
The space (X,p) is called complete if each Cauchy sequence {xn} is convergent to an element x0 E X, i.e.,
lim p(xn, x0) = 0.
It is easy to verify that a subset A of a complete metric space (X,p) is complete if and only if it is closed. A set A contained in a metric space X is called nowhere dense if the closure A of the set A does not contain any open set. A set A is said to be of the first category if it can be represented as a union of a countable family of nowhere dense sets ; otherwise, it is said to be of the second category. From the definition of the set of the category it trivially follows that a closed set of the second category contains an open set. Theorem 1.4.1 (Baire), A complete metric space (X,p) is of the second category.
Proof. Suppose that X is of the first category. Then, by definition X = W
= U Fn, where the sets F. are nowhere dense and Fn C Fn+1 Since the ft=1
set F1 is nowhere dense, there exists an open set K1 such that K1 n F1 = o and the diameter of K1
d(K1) = sup p(x, y) < 1. z,vek3
The set F2 is nowhere dense, hence there exists an open set K2 such that K2 C K1i d(K2) < 1/2 and K2 n F2 = 0.
Basic Facts on Metric Linear Spaces
19
Continuing this process, we can find by induction a sequence of open sets Kn- such that Kn C Kn+1, d(Kn) < 1/n, and
Kn n Fn = o.
(1.4.1)
Let xn e Kn. Since d(Kn)-O, {xn} is a Cauchy sequence. The space X is complete ; thus there is an element x0 E X such that {xn} tends to x0. Since
KnCKn+1,xoEKn,n=1,2,...Hence, by (1.4.1)x00Fn,n=1,2,... This implies that x0 U Fn = X and we obtain a contradiction. a=1
COROLLARY 1.4.2. Let (X,p) be a complete metric space. Let E C X be a set of the first category. Then the set CE = X\E is of the second category. Proof. X = E u CE. Suppose that the set CE is of the first category. Then X as a union of two sets of the first category is also of the first category. This contradicts Theorem 1.4.1. A metric linear space X is said to be complete if it is complete as a metric space. A closed linear subspace of a complete metric linear space is also complete.
Let (X,p) be a complete metric linear space. Let p'(x,y) be another metric on X equivalent to the metric p(x, y). Then (X,p') is not necessarily complete, as follows from
Example 1.4.2 Let X be a real line and let p(x,y) = Ix-yj. Obviously (X,p) is complete. Let p'(x, y) = jarc tan x-arc tany1. It is easy to verify that the metrics p and p' are equivalent. The space (X, p') is not complete, since the sequence {xn} = {n} is a Cauchy sequence with respect to the metric p'(x, y) but, evidently it is not convergent to any element xo a X.
The situation is different if we assume in addition that the metric p'(x,y) is invariant. Namely, the following theorem holds THEOREM 1.4.4 (Klee, 1952). Let (X, p) be a complete metric linear space. Let p'(x, y) be an invariant metric equivalent to the metric p(x, y). Then the space (X, p') is complete.
Chapter 1
20
The proof is based on the following lemmas : LEMMA 1.4.5 (Sierpiriski, 1928). Let (E,p) be a complete metric space. Let E be embedded in a metric space (E', p'). Suppose that on the set E n E' th e metrics p(x, y) and p'(x, y) are equivalent. Then E is a Ga set (i.e. it is an intersection of a countable family of open sets) in E'. Proof. Since p and p' are equivalent, for any x e E there exists a positive
number rn(x) < 1/n such that y e E, p'(x,y) < rn(x) implies p(x,y) < < 1/n. Let Un(x) _ {y e E': P'(x, y) < rn(x)}, Go
G. = U Un(x), ZEE
Go = 1 1 Gn n=1
The sets UU(x) are open. Thus the sets G,, are also open. Therefore Go is a GS set.
Evidently E C Go. It remains to show that E J Go. Let xo e Go. Then x° a Gn, n = 1, 2, ... By the definition of the sets Gn, there exist elements x,, e E such that p'(xn,xo) < r(xn). Since rn(x) < 1/n, the sequence {xn} tends to xo in the metric p'(x, y). Let e be an arbitrary positive number. Let n be a positive integer such that 2/n < e, and let ko be a positive integer such that k° < rn(xn)-p'(xn, xo).
Then, for k > ko, p'(xk, xn)
k Hence, by the definition of the number rn(xn), p (xk, xn) < l 1n. This implies that the sequence {xn} is a Cauchy sequence with respect to the metric
p(x,y). Since (E,p) is complete, there is an element x° e E such that lim p (xn, x°) = 0. The metrics p (x, y) and p'(x, y) are equivalent on the 00
set E.
Therefore xo = x° and xo e E.
Basic Facts on Metric Linear Spaces
21
LEMMA 1.4.6 (Mazur and Sternbach, 1933). Let(X, p) be a complete metric
linear space. Let the metric p(x,y) be invariant. Let X0 be a dense linear subset of X. If X0 is a G. set, then X0 = X. Go
Proof. If Xa is a Gb set, then by definition Xo = U Gn, where G. are open
sets, G. C Gn+,. Since X0 is dense in X, Gn are also dense in X. This implies that the sets X \G. are nowhere dense. Thus X\XO is the set of the first category. Then the set X0 is of the second category.
Suppose that the set X\Xo is not empty. Let x e X\XO. Since Xo is a linear subset,
x+X0 e X\Xo This leads to a contradiction, since the invariance of the metric p(x,y) implies that x+Xu is of the second category and X \Xo is of the first category. LEMMA 1.4.7 Let (X, p) be a metric linear space. Let p(x, y) be an invariant
metric. Then there is a complete metric linear space (Y,p') such that X is a dense subset in Y and the metrics p (x, y) and p'(x, y) are equivalent on X. The space Yis called the completion of the space X. Proof. We complete X in the classical way, defining Y as the space of all
sequences {xn} satysfying the Cauchy condition. We identify two sequences, {xn} and {yn}, if lim p(xn, yn) = 0. A metric in Y is defined by co
the formula
p'({xn}, {Yn}) = lim p(Xn, yn). It is well known that Y is a complete metric space. X can be embedded in Yas the set of stationary sequences x = {x, x, ...} ; X is dense in Yand on X the two metrics p (x, y) and p'(x,y) coincide. The fact that the metric p(x,y) is invariant implies that Yis a linear space and that the operations of addition and of multiplication by scalars are continuous in the metric p'(x,Y) Proof of Theorem 1.4.4. Let (Y,p') be the completion of (X,p). By Lemma 1.4.5. X is a dense Ga set in Y. Hence, by Lemma 1.4.6, X = Y.
Chapter 1
22
PROPOSITION 1.4.8. Let {xn} be a Cauchy sequence of elements of a metric
space (X,p). If it contains a subsequence {xnk} convergent to x0 a X, then {xn} also converges to x0. Proof. Let e be an arbitrary positive number. Since xnk-)-xo, there is an index ko such that for k > ko, p (xn,t, xo) < e/2. On the other hand, the sequence {x.,,} is a Cauchy sequence. Therefore, there exists a number N
such that, for n, m > N,
p(xn, xm) < 2 Putting m = nk, k > ko, we obtain p(xn, xo)
+ 2 = e.
The arbitrarines of a implies the proposition. PROPOSITION 1.4.9. Let (X, II II) be an F*-space. Let {Ei} be an arbitrary 00
fixed sequence of positive numbers such that the series
ei is convergent. i=1
If for each sequence {xi} of elements of X such that IIxiII < Fi the series w
xi is convergent, then the space (X, II II) is complete. i=1
Proof. Let {yn} be an arbitrary Cauchy sequence of elements of the space X. We can choose a subsequence {Y-k} such that IIxklI < ek, where xk
Ynk+1-Ynk The assumption implies that the series E xk is convergent k=1
to an element x e X. We shall show that {yn} tends to
In fact sk
k
i=1
X. = Ynk-Ynl tends to x. Thus ynk tends to x+yn1. By Proposi-
tion 1.4.8 {yn} tends to x+ynl. A complete F*-space is called an F-space. A complete pre-Hilbert space (Example 1.3.10) is called a Hilbert space. THEOREM 1.4.10. Let (X, II II) be an F-space. Let Y be a subspace of X.
Then the quotient space X/Y (see Section 1.1) is an F-space (i.e. it is complete).
Basic Facts on Metric Linear Spaces
23
Proof. Let {Zn} be an arbitrary sequence of elements of X/Y such that IIZnII < 1/2n. By the definition of the norm in the quotient space there are xn e Zn such that Ilxnli < 1/2n. The space X is complete, hence the series co
Y xn is convergent to an element x e X. Let Z denote the coset containing n=1
x. Then, by the definition of F-norm in the quotient space k
k
Zi--Z
-x
i=m
Therefore, the series
Z is convergent to Z, and by Proposition 1.4.9 n=1
the space X/Y is complete. PROPOSITION 1.4.11. The product (X, II II) of n F-spaces is an F-space.
Proof Let xm = (xi) be a Cauchy sequence. Then the sequences {x;'}, i = 1, ..., n also is a Cauchy sequence. Since (Xi, II Ili) are complete, there are xi e Xi such that IIxm-x{ IHO, i = 1, ..., n. Let x = (xi). Then n
IIxm_xII = I Iix'i -xi II{-*O. i=1
1.5. COMPLETE METRIC LINEAR SPACES. EXAMPLES
PROPOSITION 1.5.1. The spaces N(L(Q, E, u)) are complete. Proof. Let {xn} be a Cauchy sequence in N(L(Q, E, p)). It is easy to verify
that the sequence {xn} is a Cauchy sequence with respect to the measure
(this means that, for each a > 0,
lim p({t: Ixn(t)-xm(t)I > a}) = 0). Therefore, by the Riesz theorem, the sequence {xn} contains a subsequence {xnk} convergent almost everywhere to a measurable function x(t). Let a be an arbitrary positive number. Since the sequence {xn} is a Cauchy sequence, there is a positive integer N such that for n, m > N
PN(xn-X.) = f N(I xn(t)-xm(t)I )dl1 < 6. a
Chapter I
24
Put m = nk and let k-*oo. By the Fatou lemma, we obtain
PN(xn-x) < E. This implies that xn-x E N(L(Q, E, p). Since N(L(Q, E, p)) is linear, x c- N(L(Q, E, p)). The arbitrariness of E implies that x,-->x. PROPOSITION 1.5.2. The space M(Q, E, p) is complete.
Proof. Let {xn} be a Cauchy sequence in M(Q, E, p). Then the sequence {xn(t)} is convergent for almost all t. Let x(t) denote the limit of the sequence {xn(t)}. It is easy to verify that x(t) e M(SQ, E, p). Let E be an arbitrary positive number. Since the sequence {xn} is a Cauchy sequence, there is a positive integer N such that for n, m > N
Ixn-x.11 = esssup Ixn(t)-xm(t)I < E. LEO
Hence, when m tends to infinity, we obtain
Ilxn-x11 = esssup Ixn(t)-x(t)I < E. tES2
The arbitrariness of s implies that x.->x.
El
PROPOSITION 1.5.3. The space C(Q) is complete.
Proof. Let {X-n} be a Cauchy sequence in C(Q). This implies that at each t the sequence of scalars {xn(t)} is also a Cauchy sequence. Its limit x(t) is continuous as a limit of a uniformly convergent sequence of continuous functions. Let a be an arbitrary positive number. Since the sequence {xn} is a Cauchy sequence, there is a positive integer N such that for n, m > N
Ilxn-xmll = sup I xn(t)-xm(t)I < E. LEO
Hence, when m tends to infinity, we obtain
Ilxn-xII = sup Ixn(t)-x(t)I < The arbitrariness of s implies the proposition. PROPOSITION 1.5.4. The space C(QI Q0) is complete.
Proof. C(QIQ0) is a closed subspace of the space C(Q).
Basic racts on Metric Linear Spaces
25
PROPOSITION 1.5.5. The space eo(Q) is complete.
Proof. Let {xn} be a Cauchy sequence in Co(Q). Then {xn} is also a Cauchy sequence in each pseudonorm II Ili, i.e.,
i = 1, 2, ...
Ilxn-xmlli = 0,
lim m,n-* co
The sequence {xn(t)} is convergent for each t. Its limitx(t) is a continuous co
function on each Di, hence it is continuous on the set Q = U A and
lim Ilxn-xlli = 0,
i = 1, 2, ...
n-aco
PROPOSITION 1.5.6. The space C°°(Q) is complete.
Proof. Let {xn} be a Cauchy sequence in the space C°°(Q). Then {xn} is a Cauchy sequence in each pseudonorm II Ilk, i.e., urn m,n-
IIXn-XmIIk = 0. 4D
Let k = (0, ..., 0). This implies that the sequence {xn(t)} tends uniformly
to a continuous function x(t). In a similar way we can prove that the sequence of derivatives tends uniformly to the corresponding derivative of x(t). Thus C°°(SQ) is complete. PROPOSITION 1.5.7. The space cS(EP) is complete.
Proof. Let {xn} be a Cauchy sequence in cS (ED). Then
lim Ilxn-x.' II m,k = 0
for all m and k.
(1.5.1)
Putting m = (0, ..., 0), we infer by Proposition 1.5.6 that the sequence {xn(t)} is uniformly convergent to an infinite differentiable function together with all its derivatives.
Let a be an arbitrary positive number. Formula (1.5.1) implies that there is a positive integer N such that, for n, n' > N, IIXn-x. II a,ei
= sup (t,.... , ta) E En
I
ti k1 ... k,
P
(xn(t)-xn,(t))I JItilm< <e. ti=1
Chapter 1
26
Hence, when n' tends to infinity, we obtain a,ki
11x--x11 = sup teEP
ti
P
.k
(xn(t)-x(t)) 11 Ittlm, < S.
iT,
t=1
The arbitrariness of e implies that xn-*X. PROPOSITION 1.5.8. The spaces M(am, n) and LP(am, n) are complete.
Proof. Let {xk} = {{xn}} be a Cauchy sequence. Since sup a.,, > 0, the m
sequence {xn} converges for any fixed n to a limit xn. Write x° = xn. The sequence {xk} is a Cauchy sequence, hence it is a Cauchy sequence in each pseudonorm IIXIIm, i.e. for any s > 0, there is a positive integer K such
that for k, k' > K Ilxk-Xk'IIm
= supam,n Ixn-xn'l < e, n
(for 0
IIXk-Xk'II m = I (am, n Ixn-xn' I)P < E n
and for 1 < p < +oo 11Xn
I
lm = ( , (am, nI
xn-xn'
I )P)1/P
<e).
n
Hence, when k'-±oo we obtain IIXk-x°II < e.
The arbitrariness of a implies the proposition.
1.6. SEPARABLE SPACES
A metric space (X, p) is called separable if it contains a countable dense set. PROPOSITION 1.6.1. A metric space (X, p) is separable if and only if for an
arbitrary positive a there is a countable set E. such that for an arbitrary x e X, there is an element y e Ee such that p(x, y) < e.
Basic Facts on Metric Linear Spaces
27
Proof. Necessity. Let (X,p) be a separable space and let E be a dense countable set. Let EE = E for all e. Sufficiency. Let EE be a family of sets satisfying the property described 00
above. Then the set E = U E1l, is a dense countable set in X. n=1
COROLLARY 1.6.2. Let (X,p) be a metric space. If there are a non-countable
set A and a number S > 0 such that
p(x,y)>6
for allx,yeA,x#y,
(1.6.1)
then the space (X,p) is non-separable. Proof. Suppose that (X,p) is separable. Then by Proposition 1.6.1 there is
a countable set EE/2, such that for each x e A, there is an x e EE/2, such that p(x,z) < e/2. Thus by (1.6.1) there is a one-to-one correspondence between x and x. This leads to a contradiction since A is non-countable and EE12 is countable. PROPOSITION 1.6.3. A subset Xo of a separable metric space (X,p) is a separable metric space (X0, p ), where p' is the restriction of the metric p to X..
Proof Let e be an arbitrary positive number. By Proposition 1.6.1 there is a countable set E812 such that for any x e X there is a zz a EE12 such that
p(x,zz) < E/2. We associate with each element z E EE/2 an element y(z) e X0, such thatp(z,y(z)) < E/2, provided that such an y(z) exists. Then P (X' Y (Z.)) < AX, zz) +P (zz, Y (Z.)) < 2 +
2 =E.
Thus, by Proposition 1.6.1 Xo is separable. Let (S2, E, !e) be a measure space. The measure p is called separable if there is a countable family of sets A. e E, such that, for any set B E E of finite measure and for an arbitrary positive e, we can find a set An0 such that
p(B\A..)+ju(A..\B) < E. PROPOSITION 1.6.4. A space N(L(.Q, E, p)) is separable if and only if the measure u is separable.
Chapter 1
28
Proof. Sufficiency. Let {An} be a countable family of sets with the property described above. Let S?X be the set of all simple functions
x x(t) _ E akXAnk k=1
where, as usual, XB denotes the characteristic function of the set B, ak are
rationals in the real case and complex rational (i.e., of the form an = bn+cn i, where bn and cn are rational) in the complex case.
It is easy to verify that the set U is countable and that it is dense in N(L(Q, E. p)). Necessity. If the measure p is non-countable, then there are a noncountable family of sets {Aa} of finite measure and a positive constant 6
such that
p(AQ \Ap)+p(AI\Aa) > 6 for a # P. Let xa = XAa. Then pr,(xa, xfl) > N(1)6 for a
i9. Since the set {xa} is
non-countable, by Corollary 1.6.2 the space N(L(Q, E, p)) is not separable.
PROPOSITION 1.6.5. A space M(Q, E, p) is separable if and only if the measure p is concentrated on a finite number of atoms p, ..., pk Proof. Sufficiency. Suppose that the measure p is concentrated on a finite number of atoms pl, ..., pk. Then the space M(S2, E, p) is finite dimensional, and thus separable. Necessity. If the measure p is not concentrated of a finite number of atoms, then there is a countable family of disjoint sets {An} (n = 1, 2, ...) of positive measure. Let a = {n1, n2, ...} be a subset of the set of positive integers. Let A
= Ant V Ant v ... The family {Aa} is non-countable and for a p(Aa\A,6)+p(AB\Aa) > 0. Let xa = XAa. Then Ilxa-xxII = 1 for a =;-4 9. Therefore by Corollary 1.6.2 the space M(Q, E, p) is not separable. PROPOSITION 1.6.6. A space C(Q) is separable if and only if the topology
in the compact set Q is metrizable (i.e., it can be determined by a metric d(t, t')).
Basic Facts on Metric Linear Spaces
29
Proof. Let (.2, d) be a metric compact space. Then for each n = 1, 2, ..., there is a finite system of sets {An,k}, k = 1, ..., Kn, such that An,k n An,k' = 0
for k
k',
K. n=1
sup{d(t, t'): t, t' c- An,k} < 1/n. The family {Aa,k}, n = 1, 2, ..., k = 1, ..., Kn, is of course countable. Let X be the space of functions x(t) of the form M
X(t) = 11 amXAnm, km ,
(1.6.2)
M=1
where am are scalars and, as usual, Xy denotes the characteristic function of a set Y. Let X be the completion of X with respect to the norm jjxjj
= sup Mt)I. ten
Let W be the set of all functions of the form (1.6.2) with coefficients am either rational in the real case or complex rational in the complex case. The set ¶U is countable and it is dense in X. Thus X is separable. K.
Let x(t) e C(Q). Let xn(t) =
an,kXAn,k E 2Y, where an,k are chosen
in such a way that inf Jx(t)-an,ki < 1/n. Since the function x(t) is conteAn.k
tinuous, the sequence {xn(t)} tends uniformly to x(t). Thus it is fundamental in X. Therefore C(Q) can be considered as a subspace of the space X Thus, by Proposition 1.6.3, C(SC) is separable. Necessity. Suppose that the space C(Q) is separable. Let {xn} be a sequence dense in the unit ball K = {x: jIxjj < 1}.
For t, t' e Q, let d(t, t') _ Y Zn Ixn(t)-xn(t')I. Since Ixn(t)j < 1, n=1
d(t, t') is always finite. It is easy to verify that d(t, t') is a metric on Si. We shall show that the topology determined by this metric is equivalent to the original topology on Q.
Chapter 1
30
Let to a S2 and let E be an arbitrary positive number. Let m be a positive
integer such that 1/2m < E/4. Since the functions xn(t) are continuous, there is a neighbourhood V of the point to such that, for t e V,
xn(t)-xn(to) I < 2
(n = 1, 2, ..., m)
Then, for t e V, 00
d(t,
to)
1
n=1
Ixn(t)-xn(to)I co
xn(t)-xn(to)I + Y-L I xn(t)-xn(to)I n=n:+1
n=1
Zn
Conversely, let V be an arbitrary neighbourhood of to e D. Since S2 is compact, there is a continuous function x(t) such that Ix(t)I < 1 forte V, x(to) = 0, x(t) = 1 for t 0 V. The set {xn} is dense in the unit ball of the space C(Q), therefore there re iis xn such that Ilxn-X11 =tc-V
xn(t)-x(t)I < 4 . I
This implies that, if I xn(t) I < 3/4, then t e V and I xn(to)I < 1/4. Therefore
I xn(t)-xn(to)I <
(1.6.3)
implies that t e V. Let d(t, to) < 1/2n+1 Then (1.6.3) holds and t e V. PROPOSITION 1.6.7. A space C(QI S2o) is separable if and only if the set S2\920 is metrizable.
Proof. The proof follows the same line as the proof of Proposition 1.6.6.
PROPOSITION 1.6.8. A space e0(12) is separable if and only if the set S2 is metrizable. Proof. Sufficiency. According to the definition of the space eo(Q), the set
Basic Facts on Metric Linear Spaces
31
Sl is the union of an increasing sequence of compact sets Qm. For each m and n we can find a finite system of sets {A'"k} such that
Am n An k, = 0
for k
k',
Um
k
An,k = Slm,
sup{d(t, t'): t, t' c- An k} < 1/n. Let X be the space of functions of type (1.6.2). Let X be the completion of the space X with respect to the metric induced by the F-norm. OD
1
i=1
2i
IIxHIs, 1+IIxIIs,
where IIxik{ = sup lx(t)l. tEa,
Let Q1 be defined in the same way as in Proposition 1.6.6. The set 9C is countable and dense in X. Therefore X is separable. Then by Proposition 1.6.3. eo(Q) is separable. Necessity. Let {x.} be a dense sequence in the space e,(Q). Let for t, t' e Q
I xn(t)-xn(t')I
d(t, t') = n=1
1+Ixn(t)-xn(t')I
It is easy to verify that d(t, t') is a metric. In the same way as in the proof of Proposition 1.6.7 we can show that the metric d(t, t') induces a topology equivalent to the original one. Let (Xi, II Ili) be a sequence of F*-spaces. Let X = (Xi)(,) be the space of all sequences x = {xj x{ e X{}. The topology in X is given by a sequence of pseudonorms IIxII' = IIxiII{. Of course the space Xis an F*-space.
PROPOSITION 1.6.9. It the spaces (Xi, II Ili) are separable, then the space X = (Xi)(8) is separable.
Proof. Let 9I1 denote a dense countable set in Xi. Let 9I be the set of all sequences of the form
{a,, a,, .... a,, 0. 0....}.
Chapter 1
32
It is easy to verify that the set 91 is countable and that it is dense in the space X. Let X be a linear metric space with topology determined by a sequence of F-pseudonorms IIxjIi. Let Xo = {x: Ilxlls = 0}. Let X{ be the quotient space X/Xo. The pseudonorm IIxlk£ induces in the space Xi the F-norm IIx1Ii'
PROPOSITION 1.6.10. If all the spaces XI are separable, then the space X is separable. Proof. Let X = (Xi)(8). By Proposition 1.6.9 the space X is separable. The
space X can be identified with the subspace X0 C I of all sequences {[x]i}, where [y]i denotes the coset in X{ containing y. Obviously IlxII = II[x]4I , i.e. the topology of X0 inherited from £ and
the original topology of X coincide under this identification. Therefore X, as a subspace of a separable space, is, by Proposition 1.6.3, also separable. PROPOSITION 1.6.11. The spaces CW(S2) are separable.
Proof. Let x e C°°(Q). Then xIIk =
II a k,
X(t)I J0. al kla'..
Hence, the space Xk can be considered as a linear subset of the space C(S2). Since Q is a bounded domain in the n-dimensional Euclidean space, Q is compact. Thus by Proposition 1.6.6 the space C(Q) is separable. Hence by Proposition 1.6.10 the space C°°(Q) is separable. PROPOSITION 1.6.12. The space c5 (En) is separable.
Proof. Let x e c5 (En). Then k
IIxIIm, k
in
M
tl .. akn to
ak=
x(t)110,
0
and k
lim ti ... t".
3k1
al 3k,
x(t) = 0.
Therefore Xm, k can be considered as a linear subset of the space C(QI Q0),
where Q is the one point compactification of En and Q0 is the added
Basic Facts on Metric Linear Spaces
33
point. The space Q\Q0 is metrizable. Hence by Proposition 1.6.6. C(Q1Q0) is separable. Therefore, by Proposition 1.6.3 cS (En) is also separable. PROPOSITION 1.6.13. The spaces LP(am,n) are separable.
Proof. Elements of the type {xl, x2, ..., xn, 0, 0, ... }, where xi are rational
in the real case and complex rational in the complex case, constitute a dense countable set in LP(am,n). PROPOSITION 1.6.14. A space M(am,n) is separable provided that, for any m, there is an m' such that
lim
am,n
n-- am,, n
= 0.
Proof We construct a dense countable set in the same way as in the proof of Proposition 1.6.13.
1.7. TOPOLOGICAL LINEAR SPACES
This book deals principally with metric linear spaces but, in many cases the notion of topological linear spaces can be a very useful tool. A linear space X is called a topological linear space if it is a Hausdorff space and if the operation of addition of elements and the operation of multiplication by scalars are continuous. Since the addition is continuous, the set of neighbourhoods of the form
x+ U, where U runs over the set of neighbourhoods of 0, determines a topology in X equivalent to the original topology. Of course, each metric linear space is a topological linear space with a countable basis of neighbourhoods of 0. As follows from the Kakutani construction (see Theorem 1.1.1), if there is a countable basis of neighbourhoods of 0, then there is a metric determining a topology equivalent to the original one. Moreover, this metric is invariant. In this case the topological linear space is called metrizable. A point a is called a cluster point of a set A if, for any neighbourhood
U of the point a the intersection U n A is not empty. We say that a is a cluster point of a family of sets {A,,} if it is a cluster point of each member of the family.
Chapter 1
34
Let X be a linear topological space. A family 91 of non-void subsets of X is called fundamental, if for every two sets M, N e 91, there exists an E e 21 such that E C M n N, and for each neighbourhood of zero Uthere is aset M_ e W such that M- M C U. Example 1.7.1
Let (X,p) be a metric space. Let {xn} be a Cauchy sequence in X. The family of sets {A.n}, where An = {xn, xn+l, ...}, is fundamental. PROPOSITION 1.7.2. A fundamental family 9 has at most one cluster point.
Proof. Suppose that x and y are two cluster points of the fundamental family W. Let U be an arbitrary balanced neighbourhood of 0. Since R1 is fundamental, there is an M e 91 such that M-M C U. Since x, y are cluster points of 91, they are also cluster points of the set M. This implies
that there are xl,yl e M such that x-x1, y-yl a M. Hence
x-y = (x-x1)-(Y-YJ+(x1-Y1) e U-{- U+ U. The arbitrariness of U implies that x = y. A subset A of a topological linear space (in a particular case, the space itself) is said to be a complete set if every fundamental family of subsets of A has a cluster point a e A.
THEOREM 1.7.3. A subset E of a complete topological linear space X is complete if and only if it is closed. Proof. Sufficiency. Let 21 be an arbitrary fundamental family of subsets of the set E. Since X is complete the family 91 has a cluster point a e X. The set E is closed, and thus a e E. Necessity. Let a be a point in the closure of E. The family {(a+ U) n E}, where U runs over all neighbourhoods of 0, is a fundamental family of subsets of the set E. Since E is complete, it has a cluster point a' a E. By
Proposition 1.7.2 a fundamental family has at most one cluster point. Thus a' = a. Therefore the set Eis closed. THEOREM 1.7.4. For any topological linear space X, there is a complete topological linear space X such that X is a dense linear subset of the space X,
Basic Facts on Metric Linear Spaces
35
and the topology in X and the topology in X coincide on X. The space X is called the completion of the space X. Proof. We define the points of X as fundamental families of subsets of the
space X. We shall identify two fundamental families W and B if 0 is a cluster point of 91-0. Further steps are similar to the proof in the metric case (cf. Lemma 1.4.7).
El
Chapter 2
Linear Operators
2.1. BASIC PROPERTIES OF LINEAR OPERATORS Let (X, 11 IIx) and (Y, II IIY) be two F-spaces. We shall denote the norms
lix and II IIY by the same symbol II 11 whenever no confusion result. A mapping A transforming a linear subset DA e X into Y is called an additive operator if II
A(x+y) = -A(x)+A(y)
for all x, y e DA.
The set DA is said to be the domain of the operator A. An additive operator A is called a linear operator if
A(tx) = tA(x)
for all x E DA and all scalars t.
A linear (additive) operator is called a continuous linear (additive) operator if it is continuous. If the spaces Xand Yare linear spaces over reals, then each continuous additive operator A is a linear operator. Indeed, the additivity of the operator A implies that A(nx) = nA(x) for every integer n. !Since
A (x) = A I n)
-F A
(n
n-fold
(x)= 1 A (x). Consequently, for arbitrary rational r, we obtain that A n
n
A(rx) = rA(x). 36
Linear Operators
37
Let a be an arbitrary positive number. Let t be an arbitrary real number. The continuity of the operator A and the continuity of multiplication by scalars imply that there is a rational number r such that 11.4((t-r)x)II < 2
and
11(t-r) A(x)II <
2
Hence
IIt A(x)-A(tx)II
C 11(t-r)A(x)II+IIrA(x)-A(rx)II+IIA(rx-tx)II < e. Therefore, the arbitrariness of a implies the linearity of the operator A. Let X and Y be complete metric linear spaces. Let A be a continuous linear operator defined on a domain DA C X with the image in the space Y. Let x0 belong to the closure DA of the domain D.A. Then, by the definition of closure, there is a sequence {xn}, xn e DA, tending to x0. The sequence {xn} is obviously a Cauchy sequence. Since the operator A is continuous, the sequence {A (xn)} is also Cauchy. The space Y is com-
plete, therefore, there is a limit y e Y of the sequence {A (x.)}. Write A (x0) = y. It is obvious that A (x0) is uniquely determined in this way. We have thus extended the operator A from the domain DA onto its closure D.A. This is the reason why in the theory of continuous linear opera-
tors we shall restrict ourselves to the operators defined on the whole space X. Let X be a metric linear space. A set B C Xis said to be bounded if, for any sequence of scalars {tn} tending to 0 and for any sequence {xn} of elements of B, the sequence {tn xn} tends to 0. In other words, a set B is called bounded if, for any neighbourhood of zero U, there is a number b such that B C bU. A sequence {xn} is called bounded if the set formed by its elements is bounded.
A linear operator A mapping an F*-space X into an F*-space Y is called bounded if it maps bounded sets upon bounded sets. THEOREM 2.1.1. Let X and Y be two F*-spaces. A linear operator A mapping X into Y is bounded if and only if it is continuous. Proof. Sufficiency. Since no confusion will result, we denote the F-norms in X and in Y by the same symbol 11 II. Suppose that the operator A is
Chapter 2
38
not bounded. Then there exist a bounded set E and a positive number e such that sup IIaA(x)II > e
ZEE
for all scalars a different from 0. The set E is bounded, therefore, for an arbitrary positive 6 there is a number b such that sup IIbxjl < 6.
ZEE
This implies that, for an arbitrary positive 6, there is an element xb such that and IIA(xs)II > E. IIxnII < 6 Therefore, the operator A is not continuous. Necessity. Suppose that the operator A is not continuous. Then there are a positive number 6 and a sequence {xn} of elements of Xtending to 0 such that
IIA(xn)II>S>0. Let Xn
xn
j,IxnII
By the subadditivity of F-norm we immediately obtain IIxnII
IIxnII < 1
IXnII
+sup 0<- 1,1
II txnll
Hence x.->O implies x;,->0.
Let to = IIxnII. Evidently the sequence {tn} tends to 0. On the other hand, IItnA(xn)II = IIA(tnxn)II = IIA(xn)II > 6 > 0.
Therefore, the bounded set {x} is transformed onto an unbounded set {A (x,)}. This implies that the operator A is not bounded.
Let Bo(X--Y) denote the set of all continuous linear operators mapping X into Y. The set Bo(X-*X) we shall denote briefly by B0(X). It is easy to verify that Bo(X- .Y) is a linear set.
Linear Operators
39
Let a be a family of bounded sets in X. By B,(X->Y) we denote the set B,(X->Y) with the topology determined by neighbourhoods of the following form : U(AO, B, e) = {A a B0(X->Y): sup IIA(x)-Ao(x)II < e, B e a, ZEB
e > 0}. The space B,(X->Y) with this topology is a topological linear space, i.e. the operation of addition and multiplication by scalars are continuous. If the family a is the family of all bounded sets, then the topology generated by a is called the topology of bounded convergence. In this case
the space B,(X->Y) will be denoted briefly by B(X-*Y). Linear operators mapping X into the field of scalars will be called linear functionals. Continuous linear operators mapping X into the field of scalars will be called continuous linear functionals. Sometimes, when there is
no danger of misunderstanding, continuous linear functionals will be called briefly linear functionals or even simply functionals. The space B(X-+K), where Kis the field of scalars (i.e. the field of reals in the real case and the field of complex numbers in the complex case), is called the conjugate space to the space X. We shall denote it by X*. It may happen that X* = {0}. In this case we say that X has a trivial dual.
2.2. BANACII-STEINHAUS THEOREM FOR F-SPACES Let (X, II IIx) and (Y, II IIY) be two F*-spaces. Let X be complete (i.e. be
an F-space). Since there is no danger of misunderstanding, we shall denote the both norms by II II A family 91 of operators belonging to Bo(X-- Y) is said to be equicontinuous if for each positive a there is a positive S such that sup {IIA(x)II : A e 91, IIxii
< b} < s.
The following theorem is an extension of the Banach-Steinhaus theorem (Banach and Steinhaus, 1927) on F-spaces. THEOREM 2.2.1 (Mazur and Orlicz, 1933). Let X21 be a family of linear operators belonging to the space Bo(X--> Y). For each x e X, let the set {A(x): A e 21} be bounded. Then the family X1 is equicontinuous.
Chapter 2
40
The proof is based on the following lemma : LEMMA 2.2.2. Let (X, II II) be an F-space. Suppose that a closed set V is
absorbing, i.e. for each x e X there is a positive number a such that, for b, 0 < b < a, bx e V. Then the set V contains an open set. Proof. Since the set V is absorbing,
X= U nV. U-1
By the Baire theorem (Theorem 1.4.1) the space X is of the second category. Therefore, there is a positive integer no such that noV is of the second category. Since noV is closed, it contains an open set U. Thus V con-
tains the open set
1
no
U.
F-1
Proof of Theorem 2.2.1. Let e be an arbitrary positive number. Let
U1= n {xe X: IIA(x)II < E}. A E R[
Since the operator A is continuous, the set U1 is closed. We shall show that
U1 is an absorbing set. Indeed, let x be an arbitrary element of X. By assumption, the set {A (x): A e ¶U} is bounded. Hence there is a positive number a such that, for b, 0 < b < a, II bA (x)II < E for all A e 21. Thus bx e U1. By Lemma 2.2.2 the set U1 contains an open set U2. Let xo e U2. The
set {A(xo): A e U} is bounded. Hence there is a positive number b, 0 < b < 1 such that II bA (xo)I I < E. Thus bxo a U1. Let U = b (U2-xo)
Of course U is a neighbourhood of 0. Let x e U. Then x = by-bxo. where y e U2. Therefore IIA(x)II < IIbA(xo)II+IIbA(y)II = E+sup {IIbzII: IbI < 1, IIzII<E}.
Hence the continuity of multiplication by scalars implies the theorem. COROLLARY 2.2.3 (Mazur and Orlicz, 1933). Let X, Y be two F-spaces. Let sequence of continuous linear operators {An} be convergent at each point x to an operator A,
lim A.(x) =A (x).
Linear Operators
41
Then the operator A is a continuous linear operator. Proof. The linearity of the operator A follows trivially from the arithme-
tical rules of limits. For each x the sequence {An(x)} is convergent, and thus bounded. By Theorem 2.2.1 the family of operators {An} is equi-
continuous, i.e., for an arbitrary positive e, there is a 6 > 0 such that IIxJI < 6 implies
IIAn(x)II < 2
Let IIxII < 6. Since A (x) is convergent to A (x), there is an index no such that
IIA (x)-Ana(x)II < 2 Hence
IIA(x)II < IIA(x)-Ano(x)II+IIAn,(x)II <2+ 2 = s. Therefore, the operator A is continuous at 0. The linearity of the operator A implies that it is continuous everywhere. COROLLARY 2.2.4. If (X, II II) and (Y, II II) are F-spaces, then the space
B(X-+Y) is complete.
Proof. Let 2Y be a fundamental family of subsets of the space B(X-Y). This implies that for any x e X the family 21(x) of the sets {{A(x): A G M} ME 2i} is a fundamental family of subsets of the space Y. The space Y is complete, and thus the family 91(x) has a cluster point A0(x). Corollary 2.2.3 implies that A0(x) is a continuous linear operator. Let B be a bounded set in X and let U be a neighbourhood of 0 in Y. Let E be an arbitrary member of W. Let M C E be a member of 2Y such that, for A, Al a M, x e B,
A(x)-AI(x) e U. Then
A(x)-Ao(x) e U. The arbitrariness of B and U implies that Ao is a cluster point of M. Thus it is also a cluster point of E, and since E was an arbitrary member of 2[, Ao is a cluster point of the family W.
Chapter 2
42
In the particular case where X is an F-space the conjugate space X* is complete.
2.3. CONTINUITY OF THE INVERSE OPERATOR IN F-SPACES
,
THEOREM 2.3.1 (see Banach, 1932). Let (X, II II) and (Y, II II) be two
F-spaces. If a continuous linear operator A maps the space X onto the space Y, then the image of any open set G is open. Proof. Let U be an arbitrary neighbourhood of 0. We shall first show that
the closure A(U) of the set A(U) contains a neighbourhood of 0. Since a-b is a continuous function of its arguments, there is a neighbourhood of zero M in the space X such that M-M C U. Since M is a neighbourhood of zero,
X=UnM. n=1
Thus
Y=A(X)=UnA(M). n=1
By the Baire category theorem (Theorem 1.4.1) there is an no such that the set no A(M) contains a non-void open set V. Hence
A(U) D A(M)-A(M) ) A(M)-A(M) j no (V- V), and the set
1
n no
(V- V) is of course a neighbourhood of 0.
For any positive E we shall write XE _ {x e X: IIxJI < e}
and
YY = {y E Y: IIYII < e}.
Let so be an arbitrary positive number. Let {ei} be a sequence of pos-
itive numbers such that Z Ei < so. As we have already shown, there is
i-1
a sequence {iii} of positive numbers such that
A(Xet) ) Y,,,
i = 0,1, 2, ...
(2.3.1)
Linear Operators
43
Let y be an arbitrary element belonging to Y,,. We shall show that there is an element x e X2EO such that
A(x) = y.
(2.32.)
Formula (2.3.1) implies that there is an element xo a Xep such that
Ily-A(xo)II < '7i' i.e. y-A(xo) e Y,,1. Putting i = 1 in formula (2.3.1), we can find that there is an element xl a XE such that
Ily-A(xo)-A(xi)II < 272Repeating this reasoning, we may define a sequence X such that xn e Xen and
of elements of
n
(2.3.3)
Ily-A (z xi) II < nn+1. s=o
Since the space X is complete, the series
xn is convergent to an ele-
ment x and, moreover, IIxil < 2so. Formula (2.3.3) implies (2.3.2). Hence the image of a neighbourhood of 0 contains a neighbourhood of 0. Let G be an open subset of the space X. Let x e G. Let N be a neighbourhood of 0 such that x+N C G. Let M be a neighbourhood of 0 in Y
such that A (N) ) M. Then
A(G) D A(x+N) = A(x)+A(N) D A(x)+M. Therefore, A(G) contains a neighbourhood of each of its points, i.e. A(G) is open. THEOREM 2.3.2 (Banach, 1932). Let a continuous linear operator A map an
F-space X onto an F-space Y in a one-to-one manner. Then the inverse operator A-' is continuous. Proof. The operator A = (A-')-' maps open sets on open sets. Hence the operator A-' is continuous. COROLLARY 2.3.3. Suppose we are given in an F*-space X two different F-norms II II and II II, Suppose that the space X is complete with respect to the both F-norms. Suppose that the F-norm II I Ii is stronger than the F-norm II II. Then the two F-norms are equivalent. Proof. The operator of identity mapping (X, 11 III) into (X, II II) is con-
Chapter 2
44
tinuous. Then by Theorem 2.3.2 the inverse operator is also continuous. This implies that the F-norms 1111 and 11 ill are equivalent.
Let X be a linear space. A family of linear functionals F defined on X is called total if the fact that f(x) = 0 for all f e F implies that x = 0. COROLLARY 2.3.4. Suppose we are given in an F*-space X two different F-norms II II and II 111. Suppose that X is complete with respect to both F-norms. Suppose that there is a total family F of linear functionals which
are simultaneously continuous with respect to both F-norms. Then the F-norms II II and II III are equivalent.
Proof. We shall introduce a new norm in the space X by the formula 11x112 = IIxII+IIxIIl
and we shall show that the space X is complete with respect to the norm 11
112. Let {xn} be a fundamental sequence in the F-norm II 112. Then it is also
a fundamental sequence in the F-norms II II and II III, because the F-norm 112 is stronger than the F-norm II II and the F-norm II III. Since X is complete in II II and II J. the sequence {xn} tends to x° in the norm II II and to xl in the norm II III. Suppose that x° r x1. Since the family F is total, there 11
is a functional f e F such that f(x°) both norms implies that
f(xl). The continuity of f in the
f(x°) = lim f(xn) =.f(xl). n--.-
and this leads to a contradiction. Therefore, x° is a limit of the sequence {xn} in the norms 11 11 and 11 III. Thus it is a limit of the sequence {x"} in the norm 11 II2 Hence the space (X,II 112) is complete. By Corollary 2.3.3 the
norm II II2 is equivalent to the norm II II and to the norm II IIi Hence the norms II II and II 11, are equivalent. El
2.4. LINEAR DIMENSION AND THE EXISTENCE OF A UNIVERSAL SPACE
Let (X, II IIx) and (Y, 11 IIY) be two F*-spaces. We say that the space X and
Y are isomorphic if there is a linear operator A mapping X onto Y in
Linear Operators
45
a one-to-one manner such that the operator A and the inverse operator A-1 are continuous. We say that an F-space (X, II II x) has a linear dimension not greater than an F-space (Y, II IIr'), and we write
dimzX < dim,Y if the space X is isomorphic to a subspace of the space Y.
If dim,X < dimi Y and simultaneously dim, Y < dim,X, we say that the spaces X and Y have the same linear dimension (Banach and Mazur, 1933) and we write it as
dimiX = dim, Y.
If two F*-spaces X and Y are isomorphic, then obviously they have the same linear dimension. On the other hand, we shall see later that there are non-isomorphic spaces with the same linear dimension. We say that the linear dimension of an F*-space X is less than the linear dimension of an F*-space Y and we write
dimiX < dim, Y
if dim,X < dim,Y and dim,Y # dimtX. Let X be a family of F*-spaces. An F*-space X0 is called a universal space with respect to isomorphism, or briefly a universal space, for the family X if for every X e X there is a subspace of X0 isomorphic to X (i.e. dimjX < dimjX0). A universal space is called sometimes universal with respect to linear dimension. For each family X of F*-spaces there is a trivial universal space defined in the following way. We enumerate all the spaces belonging to X. Let A be the set of indices. Let X _ {(Xa, 11 I1a), a e A}. Let Xu be the space of generalized sequences x = {xa}, a e A, xa a Xa, such that IIxuI = Sup Ilxa IIa < +0o. aEAl
We determine the addition and the multiplication by scalars as follows {xa}+{ya} _ {xa+ya}, t{xa} = {txa}. It is easy to verify that (X.., II II) is an F-space. Space Xb is isomorphic to the following subspace of Xu
Xa= (xeX.: xa=0fora=,Z= b).
Chapter 2
46
Unfortunately the space X. obtained in this way does not necessarily belong to the family X. The problem of universality is a problem of finding a universal space belonging to the family X. Kalton (1977) has shown that there is a universal separable F-space for the family of all separable F-*spaces. We shall present his proof here. The proof is based on several notions, lemmas and propositions. To begin with, following Kalton, we shall present what is called the packing technique. This technique was introduced by Pelczynski (1969) for another purpose. Let (E, <) be a partially ordered set. A subset A C E is called a chain if it is totally ordered. A subset A C E is called a section if b < a for an element a e A implies that b e A. Let E[c] = {a a E: a < c}. If, for each c e E, the set E[c] is a finite chain, we say that E is tree-like. PROPOSITION 2.4.1 (Kalton, 1977). If E is countable and tree-like, then there is a one-to-one non-decreasing mapping of the set N of positive integers onto E. Proof. We form a sequence {an} such that each element of E occurs in the sequence infinitely many times. We shall now define by induction an increasing sequence {mn} of positive integers as follows. As m1 we take
the smallest n such that E[an] = {an}. Suppose that elements am,, ..., amk_1 are chosen. As Mk we shall take the smallest integer n such that
n > mk_I {amt, ..., amk_1, an} is a section of E, an 0 {amt, ..., amk_1}. Since E[c] are finite for all c e E, it can be proved that each c e E is in the range of a, where a(k) = amk. Let J(E) be the set of all real valued functions defined on E and vanishing everywhere except a finite number of points (the functions of this type will be called finitely supported). By T(A) we denote the set of those elements of 9(E) which have a support contained in A. By PA we shall denote the natural projection PA: 7(E)-->Y(A), PAx = xXA. By a consistent family of F-norms we mean a collection {11 tic, c e E} of F-norms such that 11
11, is an F-norm on J(E[c]),
(2.4.1.i)
if a < c, then for x e 9'(E[a]), lixHHa. _ IixIic,
(2.4.1.ii)
lltxjJc is non-decreasing for t > 0.
(2.4.1.iii)
Linear Operators
47
For a given consistent family II IIc, we define the limit F-norm II II on Y(E) as follows n
IxII =inf {
n
uj = x, ui c- 9(E[ci]), n EN}.
Iluillcf:
i=1
a=1
Observe that II II satisfies the triangle inequality
Iix+yll < IIxII+IIxII In deed, n,
x+yll = inf
It,
Ilwillci:
wi = x+y}
i=1
i=1
n,
n,
n,
q(y)+p*(x-y) > e-Bp(y)+p(x-y) e-B(p(y)+p(x-y)) > e-Bp(x) so that q(x) > e Bp(x). Thus d(p,q) = 0. We have to show that q e 9). Clearly, for x e T (N),
sup q(tx) < supp*(tx) < 1. teE
IER
Suppose that supq(tx) < b < 1. Then for any t e R there are ut and vt tER
such that tx = ut+Vt, ut a Rn, q(ut) < b and p(vt) < b. The set V = {z a Rn: g(z) < b} is compact by Lemma 2.4.6. Let W = {z e lin (Rn, x: p*(z) < b}. This set is also compact for the same reason. However, lin{x} C W+ V and we obtain a contradiction. Thus sup q(tx) = 1 for all x 0 0, and (2.4.8) holds.
Linear Operators
57
Let m > n. By condition (2.4.9') there exist em > 0 such that
if y e Rn, q(y) <em, ItI < 1, then q(tx) = Itlq(x),
(2.4.14.i)
if y e Rm, p(y) < em, I t I < 1, then p*(tx) = I tip*(x). (2.4.14.ii)
Now suppose that y e Rm and q(y) < em. Suppose that It I < 1. If y = u+v, u e Rn, q(u)+p*(v) < em, then q(tu)+p*(tv) = I tI (q(u)+P*(v)) and
q(tv) < I tI q(y)
We shall show that the strong inequality does not hold. Indeed, suppos
that ty = u+v and q(u)+p*(v) < t a(y). Then q(u) < emI tI and p*(v) < EmI tI. Let s denote the largest real number such that q(su) < em. Then
q(u) = s'em, and so s > ItI-1. Thus q(t-lu) < em and p(t-lv) < em. Hence q(t-lu)+p*(t-lv) < q(y) and this contradicts with the definition of q.
Thus, by Proposition 2.4.4, there exists a countable dimensional' ID such that, for all p e 9, (9(N),p) is isomorphic to
F*-space (Z, 11
a subspace of Z. Now let p be any F-norm on 9(N) such that( J(N),p) is a ,9F*-space. We shall show that p is equivalent to some q e 95. Without loss of generality we may assume that p(tx) > I tI p(x)
for ItI < 1.
(2.4.15)
Indeed, if p(x) does not hold (2.4.15), we can replace p by a new equivalent norm POX) ) - sup Pl(xt>1 t
Let p*(x) = (p(x))1i2. Of course
p*(tx) > ItI1i2p(x)
for Iti < 1.
(2.4.16)
Now select any strictly increasing sequence On, 1 < On, On -*2.
For any x e (N), let 1 We recall that a linear space X is called countable dimensional if X = tin {a,, } for a sequence {a,4} of linearly independent elements.
Chapter 2
58 n
p(x) = inf {T Okp*(uk): uk a Rk, u1+ ... +un = x, n e N}. k-1
By the triangle inequality we have
p(x) < p(x) < 2p(x). If x e Rn, it is easy to show that n
Okp(uk): u1+ ... +un =
p(x) = inf I
l
x}
k=1
and
p (x) < Onp (x)
Let {M{} be an increasing sequence of positive numbers. We define
q(x) = inf {p(Y)+ 00j,Mil x(i)-Y(i)I : Y e 7 (N)). i=1
It is easy to show that q(x) is an F-pseudonorm.
If P*(Mnlen) < 2n
(2.4.17)
where en = {O, ..., 0,1,0, ...}, then q(x) <j(x) < 2p*(x). Suppose that n-th place
for a sequence {xn}, q(xn)--0. Then, from the definition of q, xn can be represented as a sum xn = un+vn such that p(un)-+0 and E M{lvn(i)I-*0. i=1
If X MMIvn(i)I < 1 then jvn(i)j < M;1 for all i, and hence, for any r, we i-1 have under condition (2.4.17) vn(k)ek I + 2r
p*(vn) < p*( k=1
.
/
Hence lim sup p*(vn) < 1/2r. Thus p*(vn)-->0. Since p(un)-*0, p(xn)-*0.
n--
This shows that under condition (2.4.17) the F-norms p and q are equivalent.
59
Linear Operators
We shall now refine the selection of Mn by the following induction procedure. We define qk(x) on Rk as follows : k
qk(x) = inf {P(y)+f MiIx(i)-y(i)I: y E Rk}. r=1
Let b be chosen in such a way that for each m the set Rn+ n {x: p*(x) bo} is compact. The existence of such a bo follows from the fact that (J(N), p) is an P F*-space. Now we shall choose an increasing sequence of positive numbers {Mn} and a decreasing sequence of positive numbers {bn}, so that
P*(Mn en) < Zn ,
n = 1, 2, ...,
bo, n = 1, 2, ... ,
bn <
if u e Rn-1 and p*(u) <
(2.4.18.a)
(2.4.18.b)
bo, then qn(u) = gn_1(u),
for n > 2,
(2.4.18.c)
if u e Rn, p*(u) < bn, then gn(tu) = It I gn(u)
fortl<1.
(2.4.18d)
To start the induction pick an M1 such that (2.4.18.a) holds and 61 such that (2.4.18.b) holds. Condition (2.4.18.c) ought to be satisfied for n > 2. Condition (2.4.18.d) follows from (2.4.18.a), and the definition of q and (2.4.16).
Now suppose that M1 < M2 < ... < Mn_1 and b1 > b2 >
... > 6n-1
have been chosen. The set {x a Rn : 6n__1
Ilxlln = fix(=)I, 1=1
we may choose Mn so that (2.4.18.a) holds, and moreover if 4_1 < p*(x) < bo and IIYIIL < Mn 1 6, then p*(x-y) > Bn 1 Bn-1p*(x) and P*(Y) < 4 bo. 1
We shall now show that (2.4.18.c) holds. Let x e Rn-1 and p*(x) < 4 bo.
Chapter 2
60
Suppose that qn(x) < gn_1(x). From the definition of q, we have
qn(x) = inf{gn_1(u)+MnIIvII1n+0np*(w): u+v+w = x, u e Rn-1, v, w c- Rn}.
And by our hypothesis there are u, v, w such that x = u+v+w and qn-1(u)+MnIIvIIn+enP*(w) < q.-,(x).
(2.4.19)
Hence
MnIIvIIn+enp*(w) < q.-1(x')-q.-1(u) < qn-1(v+w).
(2.4.20)
On the other hand, q.-1(x)
Then, by the definition of qn(x), we obtain qn(x) _ EMtIx(i)I, provided i=1
p*(x) < bn, and so gn(tx) = It I qn(x) for I t I < 1 and p*(x) < bn. Thus (2.4.18.d) is also satisfied.
To conclude, we observe that if x e R", then q(x) = limq,n+n(x). Hence if p*(x) **
inf sup q(tx) = d > 0 z#0 tER
and hence q(x) = min(d-lq(x), 1) belongs to 9 and it is equivalent to q. The space (Z, II II) constructed above is countable-dimensional and it is universal for all countable-dimensional /9F*-spaces. Now we shall show
that the completion Z of the space Z is universal for all separable F*spaces. Let X be an arbitrary separable F*-space. By Lemma 2.4.5 X x l2 contains a dense countable dimensional subspace X0, being #F*-space. Thus there is a subspace Xo,1 of the space Z isomorphic to A, Since X0 is
Chapter 2
62
dense in Xx 12, the completion Xo,1 of the space X0,1 contains a subspace isomorphic to the space Xx 12. Therefore the completion Z of the space Z contains a subspace isomorphic to the space Xx 12 and it trivially implies that Z contains a subspace isomorphic to X. 2.5. LINEAR CODIMENSION AND THE EXISTENCE OF A CO-UNIVERSAL SPACE
In this section a notion in a certain sense dual to the notion of linear dimension will be considered. We say that an F-space X has the linear codimension not greater than an F-space Y,
codimlX < codiml Y,
if Y contains a subspace Yo such that the space X is isomorphic to the quotient space Y/Yo. PROPOSITION 2.5.1. Let X and Y be two F-spaces. codimlX < codimiY if and only if there is a linear continuous operator T mapping Y onto X. Proof. Necessity. By the definition of linear codimension, there is a subspace Yo of the space Y such that the space X is isomorphic to the space Y/Yo. Let To denote this isomorphism. Write T(y) = To ([y]), where [y]
denotes the coset containing y. Since To 1 is continuous, T is also continous. It is easy to observe that T maps Y onto X. Sufficiency. Let Yo = T-1(0). The transformation T induces the continuous linear operator To mapping Y/Yo onto X defined as follows To([y]) = T (y)
To is one-to-one, and hence, by the Banach theorem (Theorem 2.3.2) its inverse is continuous. Therefore X and Y/ Yo are isomorphic.
If codimlX < codiml Y and simultaneously codiml Y < codimlX, we say that the spaces X and Yhave the same linear codimension and we write it codimlX = codiml Y.
If two spaces X and Y are isomorphic, then obviously. codimlX = = codiml Y. We say that the linear codimension of an F-space X is less than the linear
Linear Operators
63
codimension of an F-space Y, codim1X
space of the space LP([O, 1] x [0, 1]) of the functions depending only on the first variab e is not isomorphic to the space LP([0,1]) (see Kalton, 1981b). Problem 2.5.1. Let (Y, II II) be an F-space. Does there exist a subspace Yo C Y such that the quotient space is an infinite dimensional separable space ?1
Let I be a family of F-spaces. We say that an F-space X. is universal for a family X with respect to the linear codimension (or briefly co-universalfor family .) if, for each X e X codimjX < codimiXX. In the same way as in the case of linear dimension we can construct a trivial co-universal space for any family X of F-spaces. Namely we enumerate all members of family X. We construct the space Xu as in the previous section. Let Xb = {{X.}: xb = 0}. The space Xu/Xb is isomorphic to the space Xb. The problem of co-universality for a family X consists in finding a space X. e I co-universal for the family X. THEOREM 2.5.2 (Kalton, 1977). There is a separable F-space which is couniversal for all separable F-spaces.
The proof is based on several notions and propositions. To begin with, we shall extend the notion of N(l) spaces to the case where N(u) depends also on n. Namely, suppose we are given a sequence {N,,(u)} of continuous
'non-decreasing functions (defined for u > 0). Suppose that N.(0) = 0 if and only if u = 0, n = 1, 2, ... Suppose that the sequence N (u) satisfies a uniform (L12) condition, i.e. there is a K > 0 (which does not depend on n) such that Nn(2u) < KNn(u). (2.5.1) 1 Added in proof: The answer is negative (Popov, 1984).
Chapter 2
64
It is easy to verify that PN(x) _
Nn(xn) n=1
is a metrizing modular on the space of all sequences. Let Nn(1) be the space of all such x that pN(x) < +oo. By Theorem 1.2.4 the space Nn(1) is linear. Moreover, the modular pN(x) induces a metric on this space such that it is
a metric linear space. In examples which will be interesting for us the functions Nn will satisfy a stronger condition than (2.5.1), namely
Nn(u+w) < Nn(u)+Nn(w).
(2.5.2)
In this case the modular pN(x) is an F-norm. It is easy to verify that Nn(l) is complete. Indeed, let {xk} _ {xk} be a fundamental sequence. Then, for each i, the sequence {xk} is also fundamental. This implies that there is a limit
i = 1, 2, ...
lim xi = xi, co
Observe that for an arbitrary e > 0 there is an index K such that, for k, k' > K 00
Nn(xn-Xn <
PN(Xk-Xk') _ n=1
Then passing, with k' to infinity, we get PN(Xk-X) < E,
where x = {x:}. The arbitrariness of a implies the completeness of the space Nn(1).
PROPOSITION 2.5.3 (Turpin, 1976). For any separable F-space (X, 11 11), there are a space Nn(1) and a continuous linear operator T such that T maps NN(l) onto X. Proof. Let {xn} be a dense sequence in X. Let Nn(u) = 1juxnJI. Let T({a{})
_
n
aix{. By the definition of Nn(1) it follows that the operator T is well r=1
determined and continuous. The proof will be complete when we show
Linear Operators
65
that T maps Nn(l) onto X. This immediately follows from the fact that each x e X can be represented in the form 00
x=IXni, i=1 1
where IlxnilI < y j. Modifying the construction of Turpin we can obtain a stronger result. PROPOSITION 2.5.4 (Bessaga, 1982, oral communication). For any separable F-space (X, I II) there is a space Nn(1) such that for any subspace Y C X there is a continuous linear operator T mapping Nn(1) onto Y. I
The proof is based on the following LEMMA 2.5.5. Let (Y,
11
11) be a separable F-space. Let -1 sup {IIYII: Y e Y}
> 6. Then there is a sequence {yn} dense in Y such that sup Iltynll > 6 for £ER
n = 1,2, .. Proof. Let y e Y be an element such that suplltyll > 26. Let {xn} be an WE
arbitrary dense sequence in Y. We put xn
Yn =
xn+
if sup II txnll > 6, tell
1y
otherwise.
It is easy to verify that yn has the requested property. Proof of Proposition 2.5.5. Without loss of generality we may assume that the norm II II is non-decreasing, i.e. IItxII < IIxII for It I < 1. Basing on Lemma 2.5.5 we can find a number 6 > 0 and a sequence {yn} C Y such that IIYnII > 6 for n = 1, 2, ... and
U tt
{tyn} = Y.
(2.5.3)
Chapter 2
66
Let the space Nn(1) be constructed as in Proposition 2.5.3. For each ym we can find xnnti such that (2.5.4)
IIYm-X.. II < 2m . Let T: N.(1) ->. Y be defined as follows 00
T({tn})
= I tnmYm M-1
Since the norm II II is non-decreasing and Ilynll < 6, inequalities (2.5.4)
imply that the operator Tis well defined and continuous. Since (2.5.3), for every y e Y there is a sequence of reals {am} such that
I 00
Y=
am ym
(2.5.5)
m=1
and
IIamymll <2m.
(2.5.6)
By (2.5.6) the sequence t = {t.}, tn =
an {0
for n = nm, otherwise
belongs to the space Nn(1) and T(t) = y. Proof of Theorem 2.5.2 (Bessaga, 1982, oral communication). By Theorem 2.4.7 there is a universal separable F-space (X, II II). Using for this space Proposition 2.5.4 we obtain a co-universal space.
For a given separable F-space (X, II II) the construction of Turpin can give two non-isomorphic spaces. Indeed, if {xn} and {yn} are two dense sequences such that inf sup II txnll > 0 and inf sup Iltynll = 0. then obn
£ER
n
teR
viously the spaces induced by the sequences {xn} and {yn} are not isomorphic.
On the other hand if {xn} and {yn} are dense sequences such that inf sup Iltxnll > 0 and inf sup Iltynll > 0, then the spaces induced by sex
tER
n
teR
Linear Operators
67
quences {xn} and {yn} are isomorphic. Indeed, there are 6> 0 and sequences of scalars {rn} and {sn} such that n = 1, 2, ... IlSnynll > 6, lrnXnll > 6,
(2.5.7)
Now we define by induction a permutation of positive integers k (n) in the following way : k(n) is the smallest positive integer different than k(1), ...
..., k(n-1) such that 1rnX.-Sk(n)yk(n) II < 2n .
(2.5.8)
Formulae (2.5.7) and (2.5.8) imply that the spaces induced by the sequences {xn} and {yn} are isomorphic.
2.6. BASES IN FFSPACES Let (X, II I) be an F-space. A sequence {en} of elements of the space X is
called a Schauder basis (Schauder, 1927) (or simply a basis) of the space X if every element x e X can be uniquely represented as the sum of a series Co
x = f tcei.
(2.6.1)
A sequence {gn} C X is called a basic sequence if it is a basis in the space Ygenerated by itself, i.e. it is a basis in Y = lin{gn}. Evidently, if an F-space has a basis, then it is separable. For any x of the form (2.6.1), let n
Pn(x) _ I it e{. i=1
THEOREM 2.6.1. The operators Pn are equicontinuous.
Proof. Let Xl be the space of all scalar sequences y = {77{} such that the W
series
niei is convergent. The arithmetical rules of the limit trivially
imply that X1 is a linear space. Let
IIyII* = supll YnteiII. n
ti=1
Chapter 2
68
The space X1 with the norm II I I is an F*-space. We shall show that it is
complete, i.e. that it is an F-space.
Suppose that a sequence {yk}e X1 is a Cauchy sequence. Let yk _ {4}. Since {yk} is a Cauchy sequence, for an arbitrary s > 0 there is a positive integer mo such that, for m, k > mo, n
Il ym-ykll * = sup n
i=1
<e.
(gym - q/ )et
(2.6.2)
Hence n-i
ifit) et
IAn -nn)enl l < II i=1
nl [ + u (gym-,J ) et
< 2s
i=1
Consequently
Jim ('2m-'))=0,
m,k- oo
n=1, 2, ...
Thus the sequences of scalars {fin } are convergent for all n. Write 27n = lim 71' . n- co
Passing with k to infinity in inequality (2.6.2), we obtain that for m > mo n
(r/i -,qt) et iI < 2e.
sup n
(2 6.3)
i=1
Let n
s"' =
and
i?m et
sn =
?It et.
t=1
t=1
Taking into account inequality (2.6.3) we obtain Il sr-snll < II
s'n-sm II+2s
for all in > mo and all n and r. Fix m1 > mo. The sequence {s"} is a Cauchy sequence. Therefore there is a number no such that, for n, r > no, I
sn'-sn' II < s.
Linear Operators
69
Hence, for n, r > no,
Ilsn-s, If < 3e. Thus the series' nt es is convergent and y = {,71} E X1.
t-1
From inequality (2.6.3) follows n
sup n
(rlm-t7,)es <2e ti=1
for m > mo. Hence the space X1 is complete. Let A be an operator mapping X1 into X defined as follows W
A(Y)=1171e{. =1
By the definition of the space X1 the operator A is well-defined on the whole space Xt. The arithmetical rules of the limit imply that the operator A is linear. Since {en} is a basis, the operator A is one-to-one and maps X, onto X. The operator A is continuous, because co
i7iet I < sup
IA(Y)jj _
n
%=1
niet
= 11Y11*-
4=1
By the Banach theorem (Theorem 2.3.2) the inverse operator A-' is also continuous. Hence n
lipn(x)Il _ 1271 es < ILYII* = 11A1(Y)!I 11
%=1
and the operators P. are equicontinuous. Let x e X be reperesented in the form (2.6.1). Let
fn(x) = tn. It is easy to see that fn are linear functionals. They are called basis functionals.
Observe that
fn(x)en = Pn(x)-Pn-,(x). Thus from Theorem 2.6.1 immediately follows
Chapter 2
70
COROLLARY 2.6.2. The basis functionals are continuous.
Suppose we are given two F-spaces X and Y. Let {en} be a basis in X and let be a basis in Y. We say that the bases {en} and {f.} are equivalent co
if the series
OD
ties is convergent if and only if the series
tifi is con-
vergent. Two basic sequences are called equivalent if they are equivalent as bases in the spaces generated by themselves. THEOREM 2.6.3. If the bases
and {f.} are equivalent, then the spaces
X and Y are isomorphic. Proof. Let T.: X->- Y be defined as follows n
..(x)= Ttiei)ttfs i=1
i=1
By Corollary 2.6.2 the operators T. are linear and continuous. The limit T(x) = lim T,,(x) exists for all x. Thus, by Theorem 2.2.3, T(x) is continuous. Since the bases are equivalent, the operator, T is one-to-one and maps X onto Y. Thus, by the Banach Theorem (Theorem 2.3.2), the inverse operator T-1 is continuous. The following theorem is, in a certain sense, converse to Theorem 2.6.1.
be a sequence of linearly independent elements in X. Let X, be the set of all elements of X THEOREM 2.6.4. Let (X, 11 11) be an F-space. Let
which can be represented in the form 00
x=Itiei. i=1
Let P-(x)
ti ei. i=1
be a sequence of linear operator defined on X. If the operators P. are equicontinuous, then the space X, is complete and the sequence {en} constitutes a basis in this space.
Linear Operators
71
Proof. Since the operators Pn are equicontinuous, each element x of X, can be expanded in a unique manner in the series co
x=Ittec. Let X2 be the space of all sequences {t{} such that the series
is convergent. Let sup n
In the same way as in the proof of Theorem 2.5.1, we can prove that 11* is an F-norm and that (X,,
is an F-space.
Observe that IIxIl < 1I{ti}JI*.
(2.6.4)
On the other hand, the equicontinuity of the operators P. implies that if x-*0 then Hence the space X2 is isomorphic to the space X. Therefore X, is an F-space. By (2.6.4) the sequence {en} constitutes a basis in X. COROLLARY 2.6.5. Let X be an F-space. A sequence of linearly independent elements {en} is a basis in X if and only if
(1) linear combination of elements {en} are dense in X, (2) the operators
Pn(x) _
ttet
are equicontinuous on the set lin{en} of all linear combinations of the set {en}.
Chapter 2
72
COROLLARY 2.6.6. Let X be an F-space with a basis {en}. Let t1i t2, ... be an
arbitrary sequence of scalars. Let p1,p2, ... be an arbitrary increasing sequence of positive integers. Let
n=1,2, ...
Ittl>0,
i=pw+1
Let Pn+i
e;,
ti et. i=Pn+1
Let X2 = {lin e;,} be the space spanned by the elements {e,,}. Then the space X2 is complete and the sequence {en} constitues a basis in X2. n
ao
ttet.
ttet. Let PP(x)
Proof. Let x e X, x
t=1
ti=1
For y e X1, let n
ao
y=
n=1
an e,
Then P (y) = Pp
n +1(y)
P,,(y) _ V at e t t=1
for all y e X1. Therefore, by Theorem 2.6.1, the
operators Pn are equicontinuous and, by Theorem 2.6.4, X2 is complete and {e;,} is a basis in X1. A basis {e;,} of the type described above is called a block basis with respect to the basis {en}. PROPOSITION 2.6.7. Let (X, II II) be an F-space with a basis {en}. Let {xn} be
a sequence of elements of X of the form 00
xk = I tk,tet
where lim tk, t = 0, i = 1, 2, ...
i=1
If {en} is an arbitrary sequence of positive numbers, then there exist an increasing sequence of indices {pn} and a subsequence {xkn} of the sequence {xk} such that Pe+1
xkn-s=P,,+1tkn, t et l'l < En.
Linear Operators
73
Proof (by induction). Let p, = 0, xkl = x1. We denote by p$ an index satisfying the inequality Ps Xk,-[
7 fff
<e1.
Suppose that the element xkti_1 and the index pn are already chosen. The hypothesis lim tk, { = 0 implies the existence of an element xkn such kyw
that P. II
Let
=1
tk"'tell < Z En.
be an index satisfying the inequality Xkn
-f tkn, i es s=1
Then obviously Pn+,
Xkn-ftk.,setI
< en.
i=Pn+l
Observe that in Proposition 2.6.7 we can additionally require that for an arbitrary given sequence {q.} there should be {pn} satisfying the conclusion of Proposition 2.6.7 such that qn < p,,. Let X be a complex F-space formed of complex valued functions (in
particular, complex sequences). We say that X is a C*-space if x = x(t) e X implies that the conjugate function x = x(t) belongs to X and the norm in the space Xdepends only on lx(t)l. Let It be the space of real-valued functions belonging to X. Since X is a C*-space, each x e Xmay be uniquely represented in the form x = x1-f-ix2,
where x', x$ a X. PROPOSITION 2.6.8. Let X be a C*-space. Let {en} be, a basis in X,. Then {en} is a basis in X.
Chapter 2
74
Proof. Since {en} in a basis in X,, there are unique expansions of elements xl and x2 with respect to this basis :
x2=I been.
x1=>anen, n=1
n=1
Hence x can be uniquely extended with respect to the basis {en} : W
X = Y cnen, n=1
p
where cn = an+ibn. We shall now give examples of bases. Example 2.6.9 The sequence
0,...,0,1,0,...,
en
n-1, 2, ...
n-th place
is a basis in the spaces c0, 1P(0 < p < +oo), (s). This basis is called a standard basis. Example 2.6.10 (Schauder, 1927) There is a basis in the space C[0,1]. It is constructed in the following manner. We define a function uk,i(t)
(0 < i < 2k, k = 0,1, ...) in the following way:
for 0 < t < Zk and
0
uk,s(t) = 2k+1(t-
i+1 < t < 1,
for 2k < t < 2 }1
?k l
for 2i+1 2k+1
-2k+1 t- i/+l 2k
i+1 2k
Each continuous function x(t) defined on the interval [0, 11 can be uniquely written in the form Go
2s-1
x(t) = a0t+al(1-t)+z, ak,iuk,i(t), k=0 i=0
Linear Operators
75
where ao = x(1), a1 = x(O) and the coefficients ak,{ are defined as follows. We draw the chord 1(t) of the arcy = x(t) through the points of the 22k1
abscissae 2k
and we define
,
2i+ 1
ak { - x
l 2i+ 1
2k+1 )
2k+1
)'
Therefore, the sequence {t, 1-t, u0,0(t), u1,0(t), u1,1(t), u2,0(t),...}
constitutes a basis in the space C[0,1]. Example 2.6.11 Suppose we are given the following system of functions on interval [0,1] : f
hn,,(t) =
for (2n)1/r
(0
1-1 < t < 2j-1 2n+ 1 2n 2j-1 j
for 2n+1 < t < -2n, elsewhere,
n=0,1,2,..., j= 1,2, ...,2n. The system of functions
is called the Haar system. It is a basis in the spaces L"[0,1] (1 < p < +00) If there is a basis in an F-space X, then obviously the space X is separable and by Corollary 2.6.2 there are continuous linear functionals different from 0 in X. In sequel linear functionals different from 0 will be called non-trivial. Let X be an F*-space. Let {xn} be a sequence of elements of X. We say that the sequence {xn} is linearly independent if for each finite system of scalars a1i ..., an the equality
a1x1+ ... +anxn = 0
(2.6.5)
a1=a2=...=an=0.
(2.6.6)
implies
Chapter 2
76
We say that sequence {xn} is topologically linearly independent if for each sequence of scalars {a1, a2, ...} the equality
alx1+ ... +anxn+ ... = 0
(2.6.7)
a1=...=an=...=0.
(2.6.8)
implies
We say that the sequence {xn} is linearly m-independent (see Labuda and Lipecki, 1982) if for each bounded sequence of scalars {an} equality (2.6.7) implies (2.6.8). We say that a sequence {xn} of elements of an F*-space X is a Hamel basis (quasi-basis, m-quasi-basis) if (1) the linear combinations of {xn} are dense in X, (2) the sequence {xn} is linearly independent (resp. topologically linearly independent, resp. linearly m-independent). Peck (1968) showed that in each separable F-space there is a quasi-basis. (For locally convex spaces it was proved by Klee (1958)). Drewnowski,
Labuda and Lipecki (1982) extended this result on topological linear spaces. In the same paper they also showed, that if a sequence {xn} is a Hamel basis in X, then there is a sequence {b1ib2, ...} such that the sequence
{blxl, blxl+b2x2i ... , blxl+ ... +bnxn, ...} is a quasi-basis. Labuda and Lipecki (1982) showed that in each separable F*-space there is an m-quasi-basis which is not a quasi-basis. A sequence {en} in an F-space X is called an M-basis sequence, if there are continuous linear functionals e* defined on E = lin{en} such that 1
e* (et)
_ 50
fori=j, for i
j,
and if x e E and e* (x) = 0, n = 1,2, ...,thenx=0. We say that a sequence {en} is equicontinuous if a*(x)en-a0 for all x e E. Kalton (see Kalton, 1979) showed that every M-basis sequence contains an equicontinuous sequence. Let X be an F-space. We say that a sequence {xn} tends weakly to x if,
for each continuous linear functional f, f(xn)->f(x). Since, in general
Linear Operators
77
there is no possibility of extension of continuous linear functionals from a subspace Y of the space X it may happen that a sequence {xn} C Y tends weakly to x in X, but does not tend weakly to x in Y. Drewnowski (1979) constructed an F-space X which, for each basis
{en}, each of its bounded block bases tends weakly to 0 in X, but no basis in Xhas the property that its bounded block basis tends weakly to 0 in Y which is a subspace spanned on that block basis.
2.7. SOLID METRIC LINEAR SPACES AND GENERAL INTEGRAL OPERATORS
Let (Q, E, p) be a a-finite measure space. Let A be a subset of the space E, p). We say that the set A is solid if u.e A and Ivi < Jul implies v e A. Let (X, II II) be an F*-space contained set-theoretically in E, p). We say that the space X is solid if there is a basis of neighbourhoods of zero consisting of solid sets. PROPOSITION 2.7.1 (Szeptycki, 1980). If (X, II II,,) is a solid F-space, then there is an F-norm II equivalent to 11 IIo such that (i) IIxMI = II Ixi II,
(ii) Hull < llvll if Jul < Ivi. Proof. Observe that if A, B are two solid sets in L°(d2, E, p), then the set. A+B is also solid. Thus the construction of the norm lI II follows from the construction of an invariant metric (Theorem 1.1.1).
We say that X is continuously imbedded in L°(S2, E, p) and we write X Cc L°(S2, E, p) if the identity mapping from X into L°(S2, E, p) is continous. PROPOSITION 2.7.2 (Luxemburg). If (X, II II) is a solid F*-space contained in L°(S2, E, p), then it is continuously imbedded, X Cc L°(S2, E, P).
Proof. (Szeptycki, 1968). Suppose that the proposition does not hold. Then there are a neighbourhood of zero U in L°(S2, E, p) of the form
U = U(E, a) = {u e L°(S2, E, p): p{t e E: I u(t) I > a} < a),
Chapter 2
78
where E is a set of finite measure and a is a positive number, and a sequence {un} C X such that un 0 U and IIun1I <
.
Zn
Let
En = {t c- E: Iun(t)I > a}. Since un 0 U, by the solidity of the space X we have µ(En) > 0. Let 00
E = U E. m=n
The measure of E is finite. Thus we can find an mn such that
n(En\E) < 2 n-1a, where E = U E.. m=n
Let CO
E'=nE;,,
E"=nE,,. n=1
n=1
Then 0" ,u(E'\E") =.u( n E,\ n=1 n0" E,) **

CO
<(En\En) <2 n=1
(2.7.1)
Since E,, is a decreasing sequence of sets and
u(E,)? u(E.)>a, we have u(E') > a. Thus by (2.7.1)
u(E") >
.
(2.7.2)
2
Let X denote the characteristic function of the set E'. Then
aX*.0
i-r
almost everywhere on Q.,°, as r--goo. And, by Proposition 2.7.8, II IKI Iur-ul Ilm-*0.
In a similar way we can show that II u,-ulln->o. Thus
Ilu.-ullm,n-0. Therefore u; tends to u in DR. In this way we have shown that each Cauchy sequence contains a convergent sequence. This implies the completeness of DR (cf. Proposition 1.4.8).
THEOREM 2.7.10 (Aronszajn and Szeptycki, 1966). The operator K is a continuous linear operator from DR into L°(Q, E, ,u). Proof. IIKuIIm < II IKI Iulllm < llullm,n
for all n.
THEOREM 2.7.11 (Aronszajn and Szeptycki, 1966 ; see also Banach, 1931).
Suppose that X C c L°(Q1, Ei, p1) is a solid F-space and suppose that X C DR. Then the integral operator K maps X into L°(Q, E, lC) is a continuous way.
Proof. By Theorem 2.7.10 it is enough to show that the identity is a continuous mapping from (X, II II) into (Dg, II IIDH), where II IIDE is the F-norm induced in the space Dg by-the sequence of pseudonorms II Ilm,n (see Section 1.3). Accordingly, we shall introduce a new norm on X, namely Ilxlll = IIkII+IIkIID"
Linear Operators
87
and we shall show that the space X is complete with respect to this new norm. Indeed, let {un} be a Cauchy sequence in (X, II II1) Then it is a Cauchy sequence in (X, II 11) and in (Dg, 11 II Dx) Since both spaces are complete,
the sequence {un} has limits in both of them. Let u° be the limit in X and let ul be the limit in D. Since X C c L°(521, El, p1) and DR C c L°(521, El, pl), the sequence {un} is convergent in the space L°(521, E1, pl) simultaneously to u° and to ul. Thus u° = ul. Hence IIun-u°II1 =
Therefore (X, II IIJ is complete, and by Corollary 2.3.3 the two norms are
equivalent. This implies the continuity of K. THEOREM 2.7.12 (Aronszajn and Szeptycki, 1966). Let (X, II IIx) C c L°(521, El, pl) and (Y, II IIY) C, L°(52, E, p) be given F-spaces. If
X C Dg and KX C Y, then the operator K: X-+ Y is continuous. Proof. We introduce a new norm on X, IIxII = IIxIIx+IIK(x)IIY,
and, in the same way as in the proof of Theorem 2.7.11, we show that II) is complete. We take a fundamental sequence {un} in (X,11 II). It is also fundamental in (X,11 IIx) Thus there is a u e X such that IIun-ullx->0. (X,11
By Theorem 2.7.11 IIun-ullDx-+0 and by Theorem 2.7.10 Ku.->Ku in L°(52, E, p). Suppose that IIKu.-vlly->0. Since Y C, L°(52, E, p), Ku.->v in L°(52, E, p). This implies that Ku = v. Hence (X, II II) is complete. Thus, by Corollary 2.3.3, the norms IIxII and IIxIIx are equivalent. This implies the continuity of Ku. PROPOSITION 2.7.13. (Aronszajn and Szeptycki, 1966). Let K be an integral transformation. Then (Dg)0 = D.K.
Proof. Let f e Dg. Let Eg\,e. Thus
p(Dn,l r) Eg)-.0,
n = 1, 2, ...
Therefore IIfxE,Iln->o,
n = 1,2,...
Chapter 2
88
By Proposition 2.7.7, IKI IfX$,I tends to 0 almost everywhere and, by the definition of the F-pseudonorm IIXIIm,n, IfXE
Example 2.7.14
Let Q = [0,2n] with the Lebesgue measure. Let 521= Z be the set of all integers with the discrete measure #1({y}) = 1,
y = 0, +1, ±2,...
Let k(t, s) = e'. Then Dg = L(Z) = 11. Example 2.7.15 Let 52 = Z and let Q1 = [0, 2n]. Let k (t, s) = e"'. Then Dg = L1 [0, 27t]. Example 2.7.16
Let 52 = S21 = R with the standard Lebesgue measure. Let k (t, s) = e't'. Then Dg = L1(-oo, +oo).
Chapter 3
Locally Pseudoconvex and Locally Bounded Spaces
3.1. LOCALLY PSEUDOCONVEX SPACES
Let X be a metric linear space. A set A C X is said to be a starlike (starshaped) set if to C A for all t, 0 < t < 1. The modulus of concavity (Rolewicz, 1957) of a starlike set A is defined by
c(A) = inf {s > 0: A+A C sA}, with the convention that the infimum of empty set is equal to -boo. A starlike set A with a finite modulus of concavity, c(A) <+oo, is called pseudoconvex. PROPOSITION 3.1.1. Let A be an open pseudoconvex set. Then
A+A C c(A)A. (3.1.1) Proof. Suppose that (3.1.1) does not hold. Then there is an x 0 c(A)A such that x e A+A. Since the set A+A is open, there is an r > 1 such that rx 0 A+A. Then
A+A ,
c(A) A,
because x 0 c(A)A. Since r > 1, we obtain a contradiction of the defi0 nition of the modulus of concavity. Observe that we always have c(A) > 2.
A set A is said to be convex if x, y e A, a, b > 0, a+b = 1 imply ax+by c- A.
Of course, for each convex set A the modulus of concavity c(A) of the set A is equal to 2. The condition c(A) = 2 need not imply convexity, but the following proposition holds : 89
Chapter 3
90
PROPOSITION 3.1.2. Let A be an open starlike set. If c(A) = 2, then the set A is convex.
Proof. Let x,y e A. Since A is an open starlike set, there is a t > 1 such that tx, ty e A. Since c(A) = 2, A+A e 2tA. Thus tx+ty e 2tA. (x+y) e A. Therefore, for every dyadic number r, This implies that 2
rx+(1-r)y e A.
(3.1.2)
The set A is open. Then the intersection of A with the line
L = {tx+(1-t)y: t real} in open in L. Therefore there is a positive number a such that
xo = tox+(1-to)y e A
for It,I < X
x1 = t1x+(1-tl)y a A
for 11-t1I < E.
and
Applying formula (3.1.2) for x = x1 and y = x0, we find that, for every
a such that ja - rI < E,
ax+(1-a)y e A.
(3.1.3)
Since r could be an arbitrary dyadic number, (3.1.3) holds for an arbitrary real a, 0 < a < 1. Thus the set A is convex. PROPOSITION 3.1.3. Let A be a starlike closed set. If c(A) = 2, then the set A is convex.
Proof. Since the set A is closed, 2A = n sA. Thus c(A) = 2 implies s>2
that A+A C 2A. Therefore, if x, y e A, then
X +Y
y a A. This implies
(3.1.2) for every dyadic number r. Since A is closed, (3.1.3) holds. A metric linear space X is called locally pseudoconvex if there is a basis which are pseudoconvex. If moreover of neighbourhoods of zero {
c(U.) < 211p, we say that the space X is locally p-convex (see Turpin, 1966; Simmons, 1964; 2elazko, 1965). THEOREM 3.1.4. Let X be a locally pseudoconvex space. Then there is F-pseudonorms {11 jjn}, i.e. such that a sequence of IItxII" = ItIP"IIxIJn,
(3.1.4)
Locally Pseudoconvex and Locally Bounded Spaces
91
determining a topology equivalent to the original one. If the space X is locally p-convex, we can assume pn = p (n = 1, 2, ...). Proof. Let { Un}be a basis of pseudoconvex neighbourhoods of 0. Without
loss of generality we may assume that the sets U. are balanced (cf. Section 1.1). From the definition of pseudoconvexity it follows that there are positive numbers sn such that Un+Un C snUn. Let Un(24) = s°nUn
(q = 0, ±1, ±2, ...). e
For every dyadic number r > 0, -r =
at 2t, where at is equal either to t=s
0 or to 1, we put Un(r) = asUn(2s)+ ... +atUn(2t).
In the same way as in the proof of Theorem 1.1.1, we show that UU(rl+r2) ) Un(r1)+ Un(r2)
and Un(r) are balanced. Moreover, the special form of UU(r) implies Un(24r) = sn UU(r).
Let IIxIIn = inf {r > 0: x e Un(r)}.
The properties of the sets UU(r) imply the following properties of IIxIIn:
(1) Ilx+ylln < IIxjIn+IIyJIn (the triangle inequality), (2) Ilaxlln = IIxIIn for all a, Ial = 1, (3) llsn1xlln = 2°IIxIIn
Let
log 2 pnlogsn. Let IIxIIn = sup t>o
Iltell n t
By (3) IIxIIn is well determined and finite since
114. =Sup t>o
Iltxlln
tn"
= 1\t\d. sup
IItxIIn
tn.
Chapter 3
92
We shall show that IIxIIn are F-pseudonorms. Indeed, if lal = 1, then
= sup
Iltaxlln tn"
llaxlln = sup t>0
9>0
IItxIIn tP..
= Ilxlln
and (n2) holds. Let x,y E X. Then = Sup IIt(x+Y)Iin IIx+Ylln = sup llt(x+Y)Iin tP,. tP.. 1-t_
t>O
sup
IItxIIn tPn
1ACt<_8n
+ sup
IItxIIn
t P,
(IIxIIn+lIYlln
and (n3) holds. Observe that IItxIIn = sup
Ilrtxlln
= sup Ilrltlxlln It1P" r>o (rltl)P"
r>o ItIP"
= I t IP" sup 8>0
I ISXI In
Si"'
= I t I P° IIxIIn.
Hence IIxII is p.-homogeneous. This implies (n4)--(n6).
Now we shall prove that the system of pseudonorms {Ilxlln} yields a topology equivalent to the original one. Indeed, from the definition of IIxIIn+
Ilxlln < IIxIIn+
in other words, {x: kiln < r} C Un(r).
(3.1.5)
On the other hand, the sets Un(r) are starlike. Therefore, the pseudonorms IIxIIn are non-decreasing, which means that Iltxll,, are non-decreas-
ing functions of the positive argument t for all n and x e X. Then
Ilxn = Sup t>0
Itx= tPn
sup 1_< t<_ e,.
Iltxll> tPn
< IlsnXln = 2 llxlln
in other words
UU(r) C {x: kiln < 2 r} .
(3.1.6)
Formulae (3.1.5) and (3.1.6) imply the first part of the theorem.
If the space X is locally p-convex, then by the definition there is a basis of neighbourhoods of 0 {Un} such that c(Un) <21/P. Since Un are open it implies that Un + Un C 21/P Un.
Locally Pseudoconvex and Locally Bounded Spaces
93
Putting s = 2112' and repeating the construction above, we obtain a system of p-homogeneous pseudonorms determining a topology equivalent to the original one. PROPOSITION 3.1.5. Let X be a locally pseudoconvex space with a topology F-pseudonorms {II IIn}. A set given by a sequence of
A C X is bounded if and only if
an = sup {IIxIIn: x e A} <+oo.
(3.1.7)
Proof. If a set A is bounded, then for each n = 1, 2, ... there is a t > 0 such that C {x: IIxII" < 1}. Thus an < -. to
Conversely, let U be an arbitrary neighbourhood of 0. Then there are e > 0 and n such that
{x: IIxII < e} C U. By (3.1.7) e
an
Ac U.
El
A topological linear space X is called locally convex if there is a basis
of convex neighbourhoods of 0. The construction given in Theorem 3.1.4 leads to the fact that if X is a locally convex metric linear space, then the topology could be determined by a sequence of homogeneous (i.e. 1-homogeneous) F-pseudonorms. Homogeneous F-pseudonorms will be called briefly pseudonorms whenever no confusion results. Locally convex metric linear spaces are called Bo spaces. As we shall see later, for locally convex spaces there is also another much simpler construction of a sequence of homogeneous pseudonorms determining the topology. A complete B,-space is called a B0-space.
Let X be an F*-space. If the topology in X can be determined by a sequence of
pseudonorms, then the space X is locally
Chapter 3
94
pseudoconvex. In the particular case where the pseudonorms are homo-
geneous, the space X is locally convex. Indeed, the sets
1 Kn, where m
Kn = {x: Jjxjjn < 1},
constitute a basis of neighbourhoods of 0. The modulus of concavity of the set Kn is not greater than 21". Indeed, if x, y e Kn, then IIx+yI In <2. Hence
x+Y 2P"
n
2,
i.e. x+y e 2P"Kn. Thus Kn+K,, C 2P".,,. In the particular case where the pseudonorms are homogeneous the sets K. are convex. It is easy to verify that the following spaces are B,-spaces: LP(Q,2,p) for 1 *

*
We say that a set A in a metric linear space X is absolutely p-convex,
0 *

*
Z-
L'
Fig. 3.1.1
If a set A is absolutely p-convex, then it is pseudoconvex and its modulus of concavity is estimated by the following formula c(A) < 21/P. Indeed, from the definition of the absolute p-convexity it follows that
if x,y e A, then 21 P e A and
Locally Pseudoconvex and Locally Bounded Spaces
95
A+Ac211"A. On the other hand, there are pseudoconvex sets which are not absolutely p-convex for any p. For example, the following open set on the plane
{(x,y):0<x<2,0*

*
c(X) = c(Y). Proof. Let T be a linear operator mapping X into Y. If A+A C sA,
then T(A)+T(A) C sT(A). Therefore c(T(A)) < c(A). If T is an isomorphism, then Tand T-1 map open bounded sets onto open bounded sets. Thus
c(Y) = c(T(X)) < c(X) = c(T-1(Y)) < c(Y). Let (X, II II).be a locally bounded space. Let 11 11 be a p-homogeneous
norm. As a trivial consequence of Proposition 3.1.5 we find that a set A C X is bounded if and only if
sup lIxll <+00. zed PROPOSITION 3.2.4. Let (X, II II) be a locally bounded space. Let Y be
a subspace of the space X. Then c(Y) < c(X). Proof. Without loss of generality we may assume that the norm II II is p-homogeneous. Let A be an open set in Y. Let
B = U {x a X: IIx-yll < inf {IIy-zII : z e Y\A}} The set B is open and bounded. Moreover,
A= Yt B.
Chapter 3
98
Thus
A+A=(BnY)+(BnY)C(B+B)nYCc(B)BnY=c(B)A. Hence c(A) < c(B) and c(Y) < c(X). PROPOSITION 3.2.5. Let (X, II II) and (Y, II 11y) be two complete locally
bounded spaces. Let T be a continuous linear operator mapping X onto Y. Then
c(Y) < c(X). Proof. Without loss of generality we can assume that II II is p-homogeneous. Let A be an arbitrary open bounded starlike set in Y. Since T induces a one-to-one operator T mapping X/ker onto Y, by Theorems 2.3.2 and 2.1.1 RA = sup inf {IIxII : x e T-1(y)} <+oo. yEA
Let
B = T-1(A) n {x: IIxII < 2RA} .
It is easy to see that the set B is an open bounded starlike set and T(B) = A. Then c(A) = c(TB)) < c(B). Therefore c(Y) < c(X). COROLLARY 3.2.6. Let (X, II
II) be a locally bounded space. Let Y be
a subspace of the space X. Then
c(X/Y) < c(X). Proof. Basing ourselves on Lemma 1.4.7 we can assume without loss of generality that X is complete. Thus we use Proposition 3.2.5. As an immediate consequence of Propositions 3.2.3, 3.2.4 and Corollary 3.2.6 we obtain THEOREM 3.2.7. Let X and Y be two locally bounded spaces.
If dime X < dime Y, then
c(X) < c(Y). If dimjX = dime Y, then c(X) = c(Y).
Locally Pseudoconvex and Locally Bounded Spaces
99
THEOREM 3.2.8. Let X and Y be two complete locally bounded spaces. If codimjX < codim= Y, then c(X) < c(Y). If codimlX = codim: Y,
then c(X) = c(Y). THEOREM 3.2.9 (Kalton, 1977). There is a separable locally bounded complete space (X, 11 11) with an a-homogeneous norm II which is uniII
versal for all separable locally bounded spaces with a-homogeneous norms.
Proof: Let 9 be the set of all a-homogeneous norms defined on 9(N) (for the notation see Section 2.4). The set 9 satisfies conditions (2.4.3.i) and (2.4.3.ii). Condition (2.4.3.i) is obvious. For the verification of (2.4.3.ii) let us observe that putting 3 = d(Ja(p),q) we have e-Bp (x) < -q (x) < eep (x)
for x E R. Let
q(x) = inf {9 y)+e°p(x-y): y E Rn}. For x E Rn we have q(x) = q(x) and for all x E 9(N) we have
e Bp(x) < q(x) < eep(x) Hence (2.4.3.ii) holds.
Let {xn} be a sequence of linearly independent elements such that lin{xn} is dense in X. Then lin{xn} is isomorphic to (9(N),p) for a certain
p. Now we construct the space (9(E), II II) as in Section 2.4. For each a c- E we denote by ea an element of 9(E) such that
for x = a, for x # a. Choose o : N-*E to satisfy Proposition 2.4.1 and write wn = e 1
e. (x) _
{0
Then, by Proposition 2.4.4, for any e > 0 there is an increasing sequence of indices (nk) such that for any m in
(1-e)p( Observe that II the theorem.
=1
II
in
taxi)
I < (I +6)p
(3.2.1)
is a-homogeneous. Since lin {xn} = X (3.2.1) implies
Chapter 3
100
Using a different method Banach and Mazur (see Banach, 1932) showed that the space C[0,1] is universal for all separable Banach spaces, hence also for 1-homogeneous F-spaces.
From Theorem 3.2.9 follows the existence of universal spaces for separable locally p-convex (and locally pseudoconvex) spaces. To show this fact we introduce the following notion. Let (Xi, II Ili) be a sequence of F*-spaces. By (Xi)(,) we denote the space of all sequences x = {xi}, xt e Xi. The topology in (Xi)(,) is defined by F-pseudonorms n
Ilxlln =,' Ilxills In the case where all Xi are identical, Xi = X we denote briefly (Xi)(,) by (X)(8).
It is easy to verify the following facts. (Xi)(,) is an F*-space. If all spaces (Xi, II Ili) are complete, then the space (Xi)(,) is also complete. If all spaces (Xi, II Ili) are locally p-convex (locally pseudoconvex), then (Xi)() is also locally p-convex (locally pseudoconvex).
THEOREM 3.2.10. There is a separable locally p-convex space, which is universal for all separable locally p-convex spaces. Proof. Let (XP, II II) be a separable locally bounded space with a p-ho-
mogeneous norm universal for all separable locally bounded spaces with p-homogeneous norms. Let Xi = XP, i = 1, 2, ... The space (XP)(,) is a separable locally p-convex space. We shall show that it is universal for all separable locally p-convex spaces. Indeed, let X be an arbitrary separable locally p-convex space. By Theorem 3.1.4 there is a sequence of p-homogeneous pseudonorms {II IIn} determining a topology equivalent to the original one. Let
Xn,o = {x: kiln = 0} and let Xn = X/X,,, be the quotient space. The pseudonorm Ilxlln induces a p-homogeneous norm in Xn. Since no misunderstanding can arise, we U
shall denote this induced norm also by II IIn. The space (Xn, II a locally bouded space with a p-homogeneous norm II Let X = (X")(.).
IIn
IIn) is
Locally Pseudoconvex and Locally Bounded Spaces
101
The space X is locally p-convex. The space X is isomorphic to a subspace of the space X consisiting of elements where [x] is the coset in X. induced by x. On the other hand by the universality of XP, X is isomorphic to a subspace of the space (XP)(8).
Basing themselves on the conclusion of Banach and Mazur (1933), that C[0,1] is universal for all separable Banach spaces, Mazur and Orlicz (1948) showed that C(-oo,+oo) is universal for all separable Bo spaces.
THEOREM 3.2.11. There is a separable locally pseudoconvex space which is universal for all separable locally pseudoconvex spaces. Proof. Let (XP, II II) denote a separable locally bounded space with a p-homogeneous norm, being universal for all separable locally bounded spaces with p-homogeneous norms. Let {pi} be a sequence tending to 0,
for example pi = 1/i. The space (X1')(8) is the required universal space. It is easily seen to be a separable locally pseudoconvex space. Let X be an arbitrary pseudoconvex space. Without loss of generality we may assume that the topology in X is determined by a sequence of pi-homogeneous pseudonorms {II IIi}. In the same way as in the proof of Theorem 3.2.10 we define Xi and X. The space X is locally pseudo-
convex. The rest of the proof is the same. THEOREM 3.3.12 (Shapiro, 1969; Stiles, 1970; for Banach spaces see Banach, 1932). Every separable locally bounded space X with a p-homogeneous norm II II is an image of lP by a continuous linear operator A.
Proof. The proof proceeds in the same way as the proof of Proposition 2.5.3. It is only necessary to observe that in this case the space N (1) is just the space 1P. THEOREM 3.2.13. Let (X, II IIx) and (Y, II IIi') be two locally bounded spaces. Let II IIx be pg homogeneous and let II IIY be py-homogeneous.
A linear operator A from X into Y is continuous if and only if
IIAII = sup IIA(x)IIY <+00. UZI z,I
(3.2.2)
Chapter 3
102
Proof. The theorem is a trivial consequence of the fact that a set E is bounded if and only if sup {IIxljY: x e E} <+oo. If px = py then the definition of IIAII implies IIA(x)IIY < IIAII IIxIIx.
In this case the number IIAII is called the norm of the operator A. Let (X, II IIx) and (Y, II IIY) be two locally bounded spaces. Let II IIx be px-homogeneous and let IIY be py-homogeneous. Then we can always introduce p-homogeneous norms, p < min (px, py), in X and Y equivalent to the original norms. Indeed, the norms II
IIxIIX = (IIxJIx)P"X,
IIy1IY = (IIYIIY)P'PY
have the required properties. We shall now show that the norm IIAII is a py-homogeneous F-norm in the space B(X--*Y). It is obvious that IIAII = 0 if and only if A = 0. Let A, B e B(X-* Y). Then IIA+BII = sup IIA(x)+B(x)IIY < sup IIA(x)IIY+ sup IIB(x)IIY pxMIX<-1
IIzIIX<-1
IIxIIX<-1
= IIAII+IIBII
The original topology in B(X-->Y) is equivalent to the topology induced by the norm IIAII Indeed, for arbitrary E > 0, {A: IIAII < E} D U(O,B,e) = {A: I A(x)IIY < r for x e B},
where the set B = {x : IIxIIx < 1} is bounded. On the other hand for an arbitrary positive and an arbitrary bounded set B C X, U(O,B,E) D U(O,Kr,E)
{A:
IIAII < rT IPX
where r = sup{IIxIIx: x e B} and Kr = {x: IIxIIx < r}. If the spaces X and Y are complete, then by Corollary 2.2.4 the space
B(X-*Y) is complete, i.e. it is an F-space. In this particular case the conjugate space X is always a Banach space. Let (X, II IIx) and (Y, II IIY) be two locally bounded spaces. Let II IIx be px-homogeneous and let Ily be py-homogeneous. From the definiII
Locally Pseudoconvex and Locally Bounded Spaces
103
tion of the norm IIA II it follows that a family RI of linear operators is equicontinuous if and only if sup {IIAII: A e 91[} <+oo.
Then Theorem 2.6.1 implies THEOREM 3.2.14. Let Z be a complete locally bounded space with a p-homogeneous norm II II and with a basis {en}. Then
K = sup IIPnII <+oo, n
where, as before, w
ttet.
P. (,.Y t{ e{) i=1
L=1
The number K is called the norm of the basis {en}. As a consequence of Corollary 2.6.5. we obtain
THEOREM 3.2.15. Let (X, II II) be a locally bounded space. Let II II be p-homogeneous. A sequence {en} of linearly independent elements is a basis
in X if and only if the following two conditions are satisfied : (1) the linear combinations of {en} are dense in X,
(2) there exists a constant K > 0 such that n+m
*

*
xn,l+ ... +xn, k = nXn, and
PN(n (xn,l+ ... +xn, k)) > r. The rest of the proof is the same as in the case where
lim N(t) = +oo.
3.4. CALCULATIONS OF THE MODULUS OF CONCAVITY OF SPACES N(L (52, _Y, ,u))
We recall that a function N(u) is called convex if, for arbitrary nonnegative numbers a, b such that a+b = 1 and arbitrary arguments u1i u2, we have N(aul+bu2) < aN(u1)+bN(u2).
Chapter 3
112
THEOREM 3.4.1. Let N(u) = N0(uP), where No is a convex function and
0 *

*
Proof. Let a be an arbitrary positive number. Let K8 = {x: pN(x) < e}.
We shall show that the set K is absolutely p-convex. Indeed, let I aI P-+- I bI P = 1 and let x, y e K. Then
PN(ax+by) = f N(Iax(t)+by(t)Ddu
f N(Ial Ix(t)I+Ibj Iy(t)l)dp
n
= f No((IaI Ix(t)I+Ibl Iy(t)I)P)dp a
< f No(IaIPIx(t)IP+IbjPIy(t)IP)dp D
*

where Mo is a convex function. Let N(u) be a function such that one of conditions (3.4.2.1), (3.4.2.ii), (3.4.2.iii) holds. Then there is a p-homogeneous norm in N(L(Q,E,p)) equivalent to the original one. The following two theorems are, in a sense, inverse to Theorem 3.4.3. THEOREM 3.4.4. Suppose that the measure p is not purely atomic. If
lim inf N(u) = 0,
U- 00
(3.4.3)
UP
then the space N(L(Q,E,p)) is not locally p-convex.
Proof. By Corollary 3.3.5 we can assume that lim N(u) = +oo. By u- 0
(3.4.3) there is a sequence {un}-oo such that N(un)l/p -*oo. Let k,, be
the greatest integer such that k < N(un), k = [N(un)]. Since the measure It is not purely atomic for sufficiently small e > 0 there are disjoint
sets An,r, i = 1,2, ..., k, such that p(A,,,t) =
e
1 +kn'
Chapter 3
114
Let Xn,t = unXAn,t
Then PN(xn,t) < E.
NlN
On the other hand,
kn
1
(-k-
un
E kn
(knl/P) l+kn moo.
4=1
The arbitrariness of E implies the theorem. THEOREM 3.4.5. Let there be a positive number r and ani nfinite family o disjoint sets Aa such that
1 < p(Aa) < r
(3.4.4)
If
lim sup
u-). J
N(u) UP
= +00
(3.4.5)
then the space N(L(Q,E,p)) is not locally p-convex. Proof. Let e be an arbitrary positive number. By (3.4.4) there is a S > 0,
such that for each u, 0 < u < 6, there is an infinite family of disjoin sets {A.,,,.} such that N(u)
< (An n) < N(u).
Let Xn,u = uXAn,,
Then PN(xn,u) < E.
On the other hand, PN
X1, u+ ... +xk, u > ek k11P
)
and by (3.4.5)
lira sup sup o__
(u) N
u
EuP
N(k-1/P u)
rN(u) N(ki/P) = rN(u) (k 1IPu)P
(u )
EuP N(k-1/P u) x nosup o
+°°'
Locally Pseudoconvex and Locally Bounded Spaces
115
COROLLARY 3.4.6. If the measure y satisifes either the assumption of Theo-
rem 3.4.4. or the assumption of Theorem 3.4.5, then in the space LP(Q,.,µ), 0 < p < 1, there is a p-homogeneous norm determining a topology equivalent to the original one, and there is no q-homogeneous (q > p) norm with this property. In other words, c(LP(SQ,E,p)) = 21/p and there is a starlike bounded open set A C LP(S2,E,p) such that c(A) = 21/p. From Corollary 3.4.6 follows THEOREM 3.4.7. There is no locally bounded space universal (co-universal) for all separable locally bounded spaces.
Proof. Suppose that such a universal (co-universal) space X exists. Then, by definition, dim 1/P < dim,X,
(codimilP < codimiX). Hence, by Theorem 3.2.7 (Theorem 3.2.8) 21/P = c(lP) < c(X)
for all p, 0 < p < 1.
Thus c(X) = +oo and this contradicts to the fact that X is locally bounded. E Another consequence of the properties of spaces N(L(Q,E,p)) is PROPOSITION 3.4.8. There is a locally bounded space X such that the topology in X can be determined by a p-homogeneous norm for p, 0 __

*
but cannot be determined by a p0-homogeneous norm. In other words, c(X) = 21/p0, but for any open bounded starlike set A, c(A) > 211-'° Proof. Let h(u) be a positive decreasing continuous convex function defined on the interval [0,+0o) such that h(0) < po and lim h(u) = 0. U- M
Let N(u) = uP°-h(u).
Let p be a finite atomless measure. By Theorem 3.4.3 for any p, 0 *

*
Chapter 3
116
determining a topology equivalent to the original one. Theorem 3.4.4 implies that there is no p0-homogeneous norm with this property. For locally convex spaces Theorems 3.4.4 and 3.4.5 can be formulated in a stronger way.
THEOREM 3.4.9 (Mazur and Orlicz, 1958). Let N(L(Q,E,p)) be a locally convex space. If the measure p is not purely atomic, then the function N(u) is equivalent to a convex function at infinity. Proof. Since the space N(L(Q,E,p)) is locally convex, there is a positive
number a such that pN(xk) < e, k = 1, ...,n, implies PN
(X'+ ...
-{--X )
n
< 1.
Let q, p, n be positive integers. Let p < n. Let t be a positive real number. Since the measure p is not purely atomic, then, for sufficiently
large q, there are disjoint sets El, ..., En such that p(E{) = 1/nq, i = 1, ..., n. The smallest such q will be denoted by q0. Let forseEj, t
where j = k, k+ 1, ..., k-l+p(mod n).
Xk(S) =
otherwise.
0
Then PN(Xk) = nq pN(t) and
PN
xJ+...-f-xn) _ n
q
Hence
qn
N(t)
<E
implies /
N( I
1
t)<1.
N(
t) .n
Locally Pseudoconvex and Locally Bounded Spaces
117
The continuity of the function N(u) and the arbitrariness of p and n imply that
(0<w<1)
-N(t) < s implies
1q N(wt) < 1. Let N(t) > eqo and let q be the greatest integer less than wN(t)/e.
From the definition of q
+
eq *

P(t')
t,
=
t"
Let
P(t)
for t > T,
It
Q(t) = t
p(T) T2
for 0 < t < T
and U
M(u) = f Q(t)dt. 0
Since Q(t) is a non-decreasing function, the function M(u) is convex.
We shall show that the functions M(u) and N(u) are equivalent at infinity. To begin with, we shall prove that P(t) and N(t) are equivalent
at infinity. Indeed, P(t) > N(t) and for sufficiently large t, CN(t), hence P(t) < CN(t).
N(cot)1(0
Since P(t) is equivalent to N(t) at infinity and N(t) satisfies condition (Q, P(t) also satisfies that condition, i.e. there is a positive constant K such that for sufficiently large t, P(2t) < KP(t). The function Q (t) is non-decreasing, whence, for sufficiently large t,
M(t) < tQ(t) = P(t) and
M(t)>
f
Q(s)ds> 2
Q(2)=P(2).
qz
Hence, for sufficiently large t, P(t) < KP (+) < KM(t) and the functions M(t) and P(t) are equivalent at infinity. Therefore the functions M(u) and N(u) are equivalent at infinity. THEOREM 3.4.10 (Mazur and Orllcz, 1958). Suppose that there is a constant K > 0 and an infinite family of. disjoint sets {Ef} such that
I < p(E4) < K.
Locally Pseudoconvex and Locally Bounded Spaces
119'
If the space N(L(Q,E,p)) is locally convex, then the function N(u) is equivalent to a convex function at 0. Proof. Since the space N(L(S?,E,u)) is locally convex, there is a positive e such that pN(xk) < e, k = 1, 2, ..., n, implies pN
(xi+ ... n
Let p be an arbitrary positive integer and let for seEk+tn (i=0, 1,...,p-1), t xk(s) 0 otherwise. We get (k = 1,2, ... , n) pN(xk) < KpN(t) and
pn N(
t) < pN (xl+ ... n
n
+xn) Hence N(t) < elKK implies N(t)ln < 1lpn. Let 0 < N(t) < e/K,
and let p be chosen so that a/K(p+1) < N(t) < e/Kp. Then N(t/n) 1/pn < 2K/e N(t)/n.
Given p > 0. Choose a q > 1 such that N(t/q) < e for I t I < p. Condition (Q implies that there is a constant Do such that N(qt)
foru>1. Thus z
z
1
N( OX = f q(u)du = f q(u)du+ f q(u)du 0
0
1
2
= f q(u)du+q(1) f min(l,q(u))du 0
K+CNG(V X).
where K = f q(u)du and C = q(1). 0
On the other hand, trivially
NG(l') < N( X) Woyczynski (1970) extended this result to the case where M(A) take values in the product space L°(Q0,E0,P) x ... X L°(D°iE0,P) and f(s) is n-times
a matrix function.
3.8. UNCONDITIONAL CONVERGENCE OF SERIES
Let Q = N be the set of all positive integers. Let E be the algebra of all subsets of N. Let M be a vector measure defined on A e E taking its values in an F-space (X, Let x = M({n}). Since M is a vector
measure the series 00E x has the property that for each sequence ei n=1 co
taking values ' either 0 or 1 the series Z e xn is convergent. The series n=1
with this property are called unconditionally convergent series. Thus
Locally Pseudoconvex and Locally Bounded Spaces
153
we have shown that each vector measure on N induces unconditionally convergent series. But the converse is also true. Namely, each unconc ditionally convergent series I xn induces a vector measure by the formula n=1
M(E) = I xn. n=E
PROPOSITION 3.8.1. The measure M induces by an unconditionally con-
x is compact (and thus bounded).
vergent series n=1
Proof. Since the series
n=1
xn is unconditionally convergent, for each
e > 0 there is an index N such that for each sequence {en} taking values either 0 or 1 Co
Gl
En Xn
< C.
(3.8.1)
I
n=N+1 N
Thus the finite set
enxn, where {e1, ..., sn} runs over all finite systems n=1
taking values either 0 or 1, constitutes an a-net in the range of the measure M. THEOREM 3.8.2 (Orlicz, 1933). A series 00I xn is unconditionally convergent n=1
if and only if for each permutation p(n) of positive integers the series GO
E xv(n) is convergent. n=1
Proof. Suppose that the series
0'
Xn is unconditionally convergent. Then
n=1
for each e > 0 there is an index N such that k+m
LI EaXn
< e
n=k
for k > N, m > 0 and en taking the value either 0 or 1.
Chapter 3
154
Let K be such a positive integer that, for n > K, p(n) > N. Then for
arbitrary r > K and s > 0 r+a
4
Eixi < E,
xy(n) II = I n=r
(3.8.2)
1=p
where
p = inf {p(n): r < n < r+s}, g = sup {p(n): r < n < r+s}, if i = p(n) (r < n < r+s), 11 Et
= to
otherwise.
The arbitrariness of E implies that the series E xv(n) is convergent. n=1 00
Suppose now that the series
xn is not unconditionally convergent. n=1
This means that there are a positive number S and a sequence {En}, E taking the value either 1 or 0 and a sequence of indices {rk} such that rk+1
LI Enxn > a.
n=rk+1
Now we shall define a permutation p(n). Let m be the number of
those s., n = rk+l ,..., rk+1, which are equal to 1. Let p(rk+v) = n(v), where n (v) is such an index that En(v) is a v-th Ej equal to 1, rk < i < rk+1r
0 < v < m. The remaining indices rk < n < rk+1 we order arbitrarily. Then rk+m
I xp(n) n=rk
rk+1 Enxn>b.
n=rk+1
This implies that the series
xP(n) is not convergent. n=1 00
A measure M induced by a series E xn is L'-bounded if and only if n=1 00
for each bounded sequence of scalars {an} the series
anxn is con-
vergent. The series with this property will be called bounded multiplier convergent.
Locally Pseudoconvex and Locally Bounded Spaces
155
THEOREM 3.8.3 (Rolewicz and Ryll-Nardzewski, 1967). There exist an 00
F-space (X, II I I) and an uncoditionally convergent series
xn of elements n=1
of X which is not bounded multiplier convergent.
The proof is based on the following lemmas. LEMMA 3.8.4. Let X be a k-dimensional real space. There exists an open symmetric starlike set A in X which contains all points pl, ..., pak of the type (e1, ..., sk), where E{ equals I or 0 or -1, such that the set
Ak-1 = A+ ... +A (k-1)-fold
does not contain the unit cube
C = {(a1i ..., ak): Ia{I < 1, i = 1,2, ..., k}.
Proof. Let Ao be the union of all line intervals connecting the point 0 with the points p, ...,per. Obviously the set Ak-1 is (k-1)-dimensional. Therefore there is a positive number a such that the set (Ao+A8)k-1, where A. denotes the ball of radius E (in the Euclidean sense), has a volume less than 1. Thus the set A = Ao+Ae has the required property. LEMMA 3.8.5. There is a k-dimensional F-space (X, II
II)
such that
i = 1,2,..., 3k and there is a point p of the cube C such that IlplI > k-1. IIp{II < 1,
Proof. We construct a norm II II in X in the way described in the proof
of Theorem 1.1.1, putting U(1) = A. Since pi e A = U(1), IIptJI < 1. Furthermore, since Ak-1 = U(k- 1) does not contain the cube C, there is a point p e C such that p e U(k-1). This implies that IIpII > k-1. Proof of Theorem 3.8.3. We denote by (Xk, II IIk) the 2k dimensional space constructed in Lemma 3.8.4. Let IIxIIk = 2k IIxIIk
Let X be the space of all sequences a = {a...} such that 00
IIIaIII = I II(as--+1, ..., k=1
<+oo.
Chapter 3
156
It is easy to verify that X is an F-space with the norm III III Let
X. = (0, 0, ..., 0, 1, 0, ...) n-th place
and let {en} -be an arbitrary sequence of numbers equal either to 0 or to 1. 00
The series I En xn is convergent. Indeed, n=1
< III YO III+III Yk+l III+ ...+III Yk'III+IIIYOI I I,
EnXn
n=m
where k is the smallest integer non-greater than log2m, k' is the largest less than log2r, and 2k
Yo
=1Enxn, n=m
j = k+1, ..., k',
Enxn,
Y1 = V1
Yo = L, Enxn n=2k'+1
In virtue of Lemma 3.8.5 1
2k_1
j=k+1,...,k'
1
2j_1 1
2k' Therefore k'
1
i=k-1
n=m
2t
1
8
< 2k_2< m,
m
and the series
D
xn is uncon-
EnXn is convergent. Thus the series n=1
ditionally convergent.
Locally Pseudoconvex and Locally Bounded Spaces
157
On the other hand, as follows from Lemma 3.8.5, for each k there is a point ak a Xk, ak = (a2a+i, ..., a2k+1), such that Ia{I < 1, i = 2k+1, ...
..., 2k+1 and
II ak Ilk >
2k
(2k-1) >
2
.
Let b = an, n = 1, 2, ... The sequence {bn} is bounded and 2k+1
I bn xn n=1
=IIakIIk>
Therefore the series
2.
bnxn is not convergent. This implies that the n=1
series E x is not bounded multiplier convergent. n=1
Observe that the measure M induced by an absolutely convergent oo
series n=1
x which is not bounded multiplier convergent is bounded, but
is not L°°-bounded. Turpin (1972) constructed a non-atomic measure with this property.
3.9. INVARIANTA(X)
We shall prove the results of Turpin (1975), however restricting ourselves to the metric case. Let (X, II II) be an F-space. By A (X) we denote the set of such sequences {An} that the space X has property P({.i;,}), i.e. for any neighbourhood of zero U there is a neighbourhood of zero V such that (3.9.1)
Of course, by definition, A (X) always contains all sequences with finite support, i.e. sequences of the form A _ {2, 22, ... 9 An, 0, 0, ...}. If A A (X) contains a sequence with an infinite support, in other words : if
there is a sequence 2 = {An}, A. > 0 belonging to A(X), we say that the space X is strictly galbed.
Chapter 3
158
PROPOSITION 3.9.1. Let (X, II II) be an F*-space. Then the set A(X) is
equal to the set of all such sequences {An} that, for any bounded sequence {xn} C X, the sequence (3.9.2)
An xn }
{YN}
i=1
is bounded.
Proof. Necessity. Let {An} e A(X). Let U be an arbitrary neighbourhood of zero. By the definition of A(X) there is a neighbourhood zero V such that (3.9.1) holds.
Let {xn} be an arbitrary bounded sequence. Then there is a b > 0 such that xn e b V, n = 1, 2, ... Thus, by (3.9.1) yN e b U, N = 1, 2,... Since U is arbitrary, the sequence yN is bounded.
Sufficiency. Suppose that {An} 0 A(X). This means that there is a neighbourhood of zero U such that (3.9.1) does not hold for any neighbourhood of zero V. This implies that for an arbitrary neighbourhood of zero V and an arbitrary positive integer n ,ZnV+An+1V+ ...'# U. (3.9.3) Now we shall choose a sequence of elements {xn} and an increasing sequence of indices {nm} in such a way that I1x{II < 2m ,
M
i = nm+1, ..., nm+1 ,
(Anm+iXnm+1+ ... +Anm+ixnm+i) 0 U.
(3.9.4)
(3.9.5)
Such a choice is possible since (3.9.3) holds.
The sequence {xn} tends to 0 by (3.9.4), thus it is bounded. On the other hand by (3.9.5) the sequence {Zm} = {)nm+lXnm+1+
+Anm+lxnm+l}
is not bounded. This implies that the sequence {yN} is not bounded either.
PROPOSITION 3.9.2. If {A.} e A (X) and 0 < /en < An, n = 1,2,..., then {µn} e A(X).
Proof. It follows immediately from the definition of A(X).
Locally Pseudoconvex and Locally Bounded Spaces
159
PROPOSITION 3.9.3. Let {AJ e A(X). Let In be a sequence of non-empty finite disjoint subsets of the set of positive integers. Let
/In= f
Ai
iEI Then {µn} e 11(X). Proof. Let {xn} be an arbitrary bounded sequence. Let x,
(xn
for i e In,
0
elsewhere.
The sequence {xtn} is of course also bounded. Then the sequence N ) N {YN} n=1
/In Xn
=
X{}
n=1 iEI,
is bounded. PROPOSITION 3.9.4. Let {An}, {/ln} a 11(X) and let (n{, m{) be a double
sequence with terms not equal to one another. Then the sequence {vt}
= {A.,-p.}e11(X). Proof. Let {xi} be an arbitrary bounded sequence. Then N
N
Vi X{ _
YN = t=1 P
_ n=1
2nullm'Xt t=1
n
(3.9.6) m
where p = max n{ and the summation with respect to m is taken over 1

*
also tends to 0. Let E be an arbitrary positive number. Since course, the set A is bounded, there is an index No such that for n > No (3.10.1)
1JCnxlj < E
for all x e A. Thus for n, m, No < n < m, Ci Xi
= 1CnxjI < E,
Cn
since m
x=
Ci
C xieA. i=n+1
Necessity. Suppose that the set A is not bounded. Then there are a sequence of numbers {tn} tending to 0, a sequence of numbers {an}, lanI < 1, and a sequence of indices {nk} such that the sequence n,,+1
anxn}
{ tk
does not tend to 0.
n=n,+1
Let Cn I
tkan 0
for nk < n < nk+1, elsewhere. W
The sequence {cn} tends to 0, but the series E CnXn is not convergent. n=1
Therefore, the sequence {xn} is not a C-sequence. An F-space X is called a C-space if for any C-sequence {xn} the series X. is convergent. As an example of an F-space which is not a C-space n=1
we can consider the space co (see Example 1.3.4.a). Indeed, the standard co
basis {en} in co is a C-sequence, but the series I en is not convergent. n=1
A subspace of a C-space is also a C-space. Therefore, if an F-space X contains a subspace isomorphic to c0, then X is not a C-space.
Locally Pseudoconvex and Locally Bounded Spaces
167
Now we shall prove the following THEOREM 3.10.2 (cf. Bessaga and Pelczynski, 1958). Let X be an F-space with a basis {en}. If X is not a C-space, then X contains a subspace isomorphic to the space co.
The proof is based on the following PROPOSITION 3.10.3 (Schwartz, 1969). Let (X, II II) be an F-space such that each C-sequence tends to 0. Then X is a C-space. Proof. Suppose that X is not a C-space. Let {xn} be a C-sequence such xn is not convergent. Then there are an increasing
that the series
n=1
sequence of indices nk and a positive number 6 such that IIykII > 6, where nk+,
Xn.
Yk =
n-nk+l
The sequence {yk} is also a C-sequence, and it does not tend to 0. Proof of Theorem 3.10.2. If (X, 11 11) is not a C-space, then there is a C-seW
quence {xn} suc11 that the series Exn is not convergent. Basing ourn=1
selves on Proposition 3.10.3, we can assume without loss of generality that the sequence {xn} does not tend to 0. Let {fk} be basis functionals corresponding to the basis {ek}. Since {xn} is a C-sequence, {fk(xn)} are C-sequences for k = 1,2, ... Hence
lim fk(xn) = 0,
k = 1, 2,...
(3.10.2)
By Theorem 1.2.2 we may assume that the norm II II is non-decreasing. Since {xn} does not tend to 0 and (3.10.2) holds, we can find a subsequence {zv}, a positive number 6 and an increasing sequence of indices {np} such that
Ilzv-z 11 < 2v 6.
(3.10.3)
Chapter 3
168
where nv+1
ZP =
fn(zp)en.
n=ni+1
Corollary 2.6.6 implies that {z,} is a basic sequence. We shall show that it is equivalent to the standard basis of co. Let {tp} be a sequence of
coefficients of x belonging to Xo = lin {z,} with respect to the basis {zp}. Since IIz,II > 6/2, the sequence {tn} tends to 0. On the other hand, (Zp} is a C-sequence. Therefore, by (3.10.3), {z,} is also a C-sequence. This means that if tn->0, then the series E tpz1, is convergent. Therefore, P=1
the basis zP is equivalent to the standard basis of co. Thus, by Theorem 2.6.3, the space X0 is isomorphic to the space co. PROPOSITION 3.10.4. Let (X, II II) be a locally bounded space. with a basis
{en}. Let Y be a subspace of the space X. Suppose Y is not a C-space. Then the space Y contains a subspace Yo isomorphic to the space co. Proof. Without loss of generality we may assume that 11 11 is p-homoge-
neous. Using the same construction as in the proof of Theorem 3.10.2, we find elements {zp} and {z,}. By formula (3.10.3) and Theorem 3.2.16 the sequence {zp} is a basic sequence. The rest of the proof is the same as in Theorem 3.10.2. Let X be a non-separable F-space. If X is not a C-space, then it contain s
an infinite dimensional separable subspace X which is not a Gspace. Indeed, if X is not a C-space, then there is a C-sequence {xn} such that xn is not convergent. Let X = tin{xn}. The space X has the
the series n=1
required properties. By the classical theorem of Banach and Mazur (1933), space C[0,1] is universal for all separable Banach spaces. The space C[O, 1] has a basis (Example 2.6.10). Thus we have PROPOSITION 3.10.5 (Bessaga and Pelczynski, 1958). If X is a Banach space which is not a C-space, then X contains a subspace isomorphic to co. Now we shall prove
Locally Pseudoconvex and Locally Bounded Spaces
169
THEOREM 3.10.6 (Schwartz, 1969). The spaces LP(Q,E,p), 0 *

*
The proof is based on the following propositions. PROPOSITION 3.10.7 (Kolmogorov-Kchintchin inequality ; see also Orlicz,
1933b, 1951, 1955). Suppose that in the space L°(Q,E,µ) a C-sequence
{xn(t)} is given. Then on each set D° of finite measure the series CO
Ixn(t)I2 convergent almost everywhere. n=1
Proof(Kwapien, 1968). Let {rn(s)} _ {sign (sin 27t2ns)} be a Rademacher
system on the interval [0, 1]. Since under our hypothesis {xn(t)} is a C-sequence, the set of elements n
Y., 40
ri(s) xi(t): n = 1, 2, ..., 0 < s <
1}
i=1
is, by Proposition 3.10.1, bounded in the space L°(Q,E,p). This implies that for any positive e there is a constant C such that, for all n = 1, 2, .. .
and all s,0<s<1, ,u({t a Do: IYn,B(t)I < C}) > /"(Q°)-E.
(3.10.4)
Let n be fixed and let
At = {s: IYn,s(t)I < C}.
(3.10.5)
By the Fubini theorem and (3.10.4) we find that the set E of those t for which the Lebesgue measure IAtI of the set At is greater than 1- rE has a measure greater than p(Qo)-j/E, i.e. if
E = {t a D°: JAtd > 2-1/E }, then
p(E) > p(Do)-r Let t e E. Formula (3.10.5) implies IS
f I ri(s) xt(t)I$ds < C2. At
i=1
Chapter 3
170
Thus
f I Ixi(t)j2 -2 f ( I Re(xi(t)xj(t))ri(s) rj(s)) ds < C2. At i=1
At
(3.10.6)
1<_i<j,*

*
Locally Pseudoconvex and Locally Bounded Spaces
173
Proof. The proof is the same as the proof of Lemma 3.10.9, but in (3.10.16) the Minkowski inequality is used.
Proof of Theorem 3.10.6. The proof is a trivial consequence of Proposition 3.10.2 and Lemmas 3.10.9, 3.10.10. COROLLARY 3.10.11. The space L°[0,1] is not universal for separable F-spaces.
Proof. The space L°[0,1] is a C-space. Hence it does not contain a subspace isomorphic to c°.
The notions of C-sequences and C-spaces introduced by Schwartz (1969) are similar to what Matuszewska and Orlicz (1968) called condition (0). A sequence {xn} of elements of an F-space X is called perfectly bounded
if the set N
A°
I xi,,: pn runs over all finite increasing systems of n=1
positive integers) is bounded. An F-space X satisfies condition (0) if each perfectly bounded sequence
is unconditionally convergent. It is easy to verify that for F-spaces satisfying condition (0) Theorem 3.10.2 holds. Therefore, if an F-space X
can be imbedded into an F-space with a basis, then X is a C-space if and only if it satisifies condition (0). Matuszewska and Orlicz (1968) showed that a large class of modular spaces (in particular, spaces N(L(Q,E,p))) satisfy condition (0).
3.11. LOCALLY BOUNDED ALGEBRAS
Let X be a linear space. We say that X is an algebra if there is an operation : X x X-->X, called multiplication, which is associative x, y, z E X
(3.11.1)
Chapter 3
174
and bilinear (x1+x2) Y = x1 Y+x2 Y,
(3.11.2) (3.11.3)
for x,Y, x1, x2,Y1,Y2 e X,
(tx) y = t (x y) = x (ty)
(3.11.4)
for x, y e X, t being a scalar. For brevity we shall write x y = xy. We say that an algebra X is commutative if
xy = yx.
3.11.5)
We say that an algebra X has a unit if there is an element e e X such that
ex = xe = x.
(3.11.6)
It is easy to verify that the unit is unique. If an algebra X has a unit, then (3.11.4) automatically holds. In the sequel only commutative algebras with a unit will be considered. For brevity we shall call them simply algebras. We say that an algebra X is an F* -algebra if it is an F*-space and the operation of multiplication is continuous. We say that an F*-algebra X is locally bounded if it is a locally bounded space. The theory of locally bounded algebras was developed by Zelazko (1960, 1962, 1962b, 1963, 1965). It is a generalization of the classical theory of Banach algebras. THEOREM 3.11.1 (Zelazko, 1960). Let X be a locally bounded algebra. Then the topology in X can be determined by a p-homogeneous norm IIxII which is submultiplicative, i.e. IIxYII < 11411A.
(3.11.7)
If X is complete, then it is also complete with respect to the norm 11 11. Proof. By Theorem 3.2.1 the topology in X can be determined by a p-homogeneous norm IIxII'. Let
114 = sup
11xY11'
IIYII
Observe that IIxII is finite, since the multiplication is a continuous oper-
Locally Pseudoconvex and Locally Bounded Spaces
175
ator (cf. Theorem 3.2.13). It is easy to verify that IIxII is p-homogeneous, submultiplicative and IIeII = 1. Since
(3.11.8)
IIxII > IIxII' IIeII'
the norm IIxII is stronger than the norm IIxII
Now we shall identify x e X with a continuous linear operator Lz mapping X into itself by formula Lzy = xy. Observe that the norm operator IIL=II of the operator Lz is equal to IIxII Thus we can consider (X, II II) as a subspace of the space (B(X), II II). If {xn} tends to x in the space (X, II II'), then, by the continuity of multiplication, Lz. tends to Lz in (X, 11 11). Thus the norms II II and II II' are equivalent. Let X be an algebra. An element x e X is called invertible if there is an element x-1 e X, called inverse to x, such that
xx-1=x 1x=e.
(3.11.8)
It is easy to verify that x-1 is uniquely determined (provided it exists) = x. and that (x-1)-1
PROPOSITION 3.11.2. Let (X, II II) be a complete locally bounded algebra.
Let x be an invertible element in X. Then there is a neighbourhood of zero U such that, for y e U, the element x-y is invertible. Proof. Without loss of generality we may assume that II is p-homogeneous and submultiplicative (see Theorem 3.11.1). We can write II
OD
(x-
y)-1
(x-1y)n,
= x1(e-x1y)_1 = x-1 n=o
where the series is convergent provided IIeII < x-1 1 II
II
Let X be an algebra. A set MC X, {0} M = X is called ideal if it is a linear subset of X and for all x e X, xMC M. Let X be an F-algebra, by the continuity of multiplication we conclude that for any ideal M its closure M is also an ideal provided that M -,/= X. We say that an ideal M C X is maximal, if for any ideal M1 such that M C M, we have M = M1.
Chapter 3
176
PROPOSITION 3.11.3. Let X be a complete locally bounded algebra. Then every maximal ideal is closed.
Proof. Let M be a maximal ideal in X. By Proposition 3.11.2, e 0 M. Thus M is an ideal. Since M is maximal and M C M, M = M. PROPOSITION 3.11.4. Let X be a complete locally bounded algebra over complexes. The function (x-Ae)-1 is analytic on its domain.
Proof. Let Ao belong to the domain of the function (x-Ae)-1. This means that the element x-Aoe is invertible. Let y = (A-Ao)e. Then (x-Aoe)-y = x-Ae. Therefore (x-Ae)-1
=
((x-Aoe)-y)-1
_ (x-Aoe)-1' (x-Aoe)-'a(A-A0)'' , n=1
where the series on the right is convergent provided IA-Ao' < JK(x-Aoe)-11h-1
Thus (x-Ae)-1 is analytic. Proposition 3.11.4 implies an extension of the theorem of Mazur and Gelfand (Mazur, 1938; Gelfand, 1941). THEOREM 3.11.5 (Zelazko, 1960). Let X be a complete locally bounded field over complex numbers. Then X = {A e: A being a complex number}.
Proof. Suppose that there is an x E X which is not of the form Ae, i.e. x Ae for all A. Since X is a field (x-Ae)-1 is well defined on the whole complex plane. By Proposition 3.11.4 it is an analytic function. Observe that
lim (x-Ae)-1 = 0. A OD
Thus, by the Liouville theorem (Theorem 3.5.4), (x-Ae)-1 = 0 and we obtain a contradiction. Let X be a complete locally bounded algebra over the complex numbers. Let f be a linear functional defined on X (not necessarily continuous). We say that the functional f is a multiplicative-linear functional if
f(xy) = f(x)f(y).
(3.11.9)
Locally Pseudoconvex and Locally Bounded Spaces
177
There is a one-to-one correspondence between multiplicative-linear functionals and maximal ideals. Namely, for a given multiplicative-linear functional f the set
M,={x: f(x)=0} is an ideal. It is a maximal ideal since it has codimension 1. Since X is locally bounded, it is closed by Proposition 3.11.3. Thus the functional f is continuous. Thus we have proved PROPOSITION 3.11.6 (2elazko, 1960). In complete locally bounded algebras all multiplicative-linear functionals are continuous.
Let M be a maximal ideal in the algebra X. Then X/M is a complete locally bounded field. Therefore it is isomorphic to the field of complex numbers (see Theorem 3.11.5) and this isomorphism induces the required multiplicative-linear functional. A locally bounded algebra X is called semisimple if
{x: F(x) = 0 for all multiplicative linear functionals} = {0}. If a complete locally bounded algebra X over complex numbers is semisimple, then x e X is invertible if and only if F(x) # 0 for all multiplicative-linear functionals F. Indeed, x is invertible if and only if x does not belong to any maximal ideal. In the case of complete locally
bounded algebras this implies that F(x) # 0 for all multiplicative functionals. Now we shall present an application of locally bounded algebras to
the theory of analytic functions. For this purpose we shall present the following example of a locally bounded algebra. Example 3.11.7
Let N(u) be a non-decreasing continuous function, defined for u >' 0,
such that N(u) = 0 if and only if u = 0. We shall assume that for sufficiently small v, u
N(u+v) < N(u)+N(v)
(3.11.10)
and there is a C > 0 such that for sufficiently small u, v
N(uv) < CN(u)N(v)
(3.11.11)
Chapter 3
178
and there is a p > 0 such that N(u) = N0(up), (3.11.12) where No is a convex function in a neighbourhood of zero. Let X = N(1) be the space of all sequences x = {xo, x,, ... } such that OD
N(IxnI) < +oo.
IIXII = PN(X)
n=o
The space (X, II II) is an F-space (see Proposition 1.5.1). By Theorem
3.4.3 it is locally bounded. Now we shall introduce multiplication in X by convolution, i.e. if x = {xn}, y = {y.} then we define n
x.y = { k=0 xkyn-k J. ((
By (3.11.10) and (3.11.11) we conclude that for sufficiently small x and y (3.11.13)
IIxYII < C IIxIIIIYII
Formula (3.11.13) implies that the multiplication is continuous. Thus X is a complete locally bounded algebra.
Now we shall give examples of functions satisfying conditions (3.11.10)-(3.11.12). The simplest are functions N(u) = up, 0 < p < 1. There are also other more complicated functions. For example 0 for u = 0, N(u) _ -up logu for 0 < u < e-2/p 2p le-2 for e-2/p < U. We shall show that N(u) satisfies (3.10.10)-(3.10.12) By the definition, N(O) = 0. The function N(u) is continuous at point 0 since lim N(u) = 0,
and at point
a-2/p since
N(e-21)
= 2p-1e-2. In the interval (0, a-2/p)
it is continuous as an elementary function. The function N(u) is non-decreasing. Indeed, on the interval (0, a-2/p)
dN
_
du
because logu < -2/p.
-pup-, logu-up-'
= -up-1(P logu+1) > 0,
Locally Pseudoconvex and Locally Bounded Spaces
179
Now we shall calculate the second derivative d2N
=
(p-1)uP-2(Plogu+1)-puP-2
du2
= -uP-2(p(p-l)logu+2p-1) = -Up-2((p-1)(plogu+l)+1) < 0 for 0 < u < e-21P. Thus N(u) is concave on the interval (0, a-2/P). Now formula (3.11.11) will be shown. Suppose we are given u, v 0 < u, v < e-2/P. Then (uv)Pjloguvj = (uv)Pllogu+logvl < uPjlogujvPjlogvJ,
because Ilogu, logvl > 2/p > 2. Let N0(x) = -x2logx. It is easy to see that if x = u'12, then (3.11.14)
NO(uP/2) = 2 N(u).
We shall show that No is convex in a neighbourhod of zero. Indeed, dNo
dx
=
dx2 -
-2xlogx-x,
-21ogx-2-1 = -2logx-3
and the second derivative is greater than 0 on the interval (0, a-312). Therefore the function No is convex in a neighbourhood of 0 and (3.11.12) holds. Example 3.11.8 Let N be as in Example 3.11.7. By N±(1) we shall denote the space of all
sequences of complex numbers x = {xn}, n = ..., -2,-1,0,1,2, ... such that co
1lxii = PN(x)
N(I xnl) < +oo.
(3.11.15)
n=-ao
In a similar way as in Example 3.11.7 we can show that (N±(1), 11
11) is
Chapter 3
180
a complete locally bounded space. We introduce multiplication a by the convolution +W
xy =
{k=-aoI
xn-kYk
In a similar way as in Example 3.11.7 we can show that N±(1) is a complete locally bounded algebra.
By NF we shall denote the algebra of measurable periodic functions, with period 2n, such that the coefficients of the Fourier expansions
x(t)
n=-
xneint
belong to the space N±(1). The operations of addition and multiplication are determined as pointwise addition and multiplication. It is easy to see that the pointwise multiplication of functions in NF
induces the convolution multiplication in the space N±(1). Thus the space NF can be regarded as a complete locally bounded algebra with topology defined by the norm (3.11.15). We shall show that each multiplicative linear functional defined on algebra N. is of the form
F(x) = x(to) (3.11.16) Indeed, let z = eit. The element z is invertible in N. We shall show that IF(z)J = 1. Suppose that IF(z)l > 1. Then there is a, 0 < a < 1 such that IF(az)l = 1. Since Fis multiplicative, it follows that IF(anzn)I = 1. On the other hand, a"zn tends to 0 and this is a contradiction since each multiplicative-linear functional in NF is continuous. If IF(z)I < 1, then JF(z 1)I > 1 and we can repeat the preceding considerations. Thus IF(z)J = 1 and F(z) = eti' for a certain to, 0 < to < 27r. Since F is a multiplicative linear functional, we obtain for every polynomial n
P(z)
i=k
a{zt
(here n, k are integers not necessarily positive),
F(p(z)) = p(F(z)) = P(e'°).
Locally Pseudoconvex and Locally Bounded Spaces
181
The polynomials are dense in algebra NF and hence the functional F is of the form (3.11.16). This implies THEOREM 3.11.9. Let N be a function satisfying the condition described in Example 3.11.7. Let x(t) be a measurable periodic function, with period 2n, such that the coefficients {xn} of the Fourier expansion
x(t) = L xnein6 ri=-ao
form a sequence belonging to the space N±(1). If x(t) :i-1: 0 for all t, then the function 11x(t) can also be expanded in a Fourier series 1
x(t)
yneint n=-00
such that {yn} e N±(1).
For N(u) = u this is the classical result of Wiener. For N(u) = uP, 0 < p < 1 it was proved by 2elazko (1960). Let X be a locally bounded complete algebra over complex numbers. Let x e X. By the spectrum a(x) of x we mean the set of such complex numbers A, that (x-)e) is not invertible. By Proposition 3.11.2, the set of such complex numbers 2 that (x-2e) is invertible is open. Moreover, if A > IIxII, the element (x-1e) is also invertible. This implies that the set a(x) is bounded and closed. Hence it is compact. Let 20 e u(x). Then by the definition of a spectrum, the element (x-toe) is not invertible. Then there is a multiplicative linear functional F such that F(x-2oe) = 0, i.e. F(x) = 20. Conversely, if 20 0 a(x), then, for each multiplicative linear functional F, F(x) 2o. Thus o(x) = {F(x): F runs over all multiplicative functionals}. Let O(.1) be an analytic function defined on a domain U containing
the spectrum a(x). Let Tc U be an oriented closed smooth curve containing a (x) inside the domain surrounded by T. We shall define O(x)
2ni
f t(2)(x-)e)-ld2.
r
Chapter 3
182
The integral on the right exists since r o a(x) = 0. It is easy to verify that, for any multiplicative linear functional F,
F(O(x)) = O(F(x)).
(3.11.17)
Applying (3.11.17) to the algebra NF, we obtain THEOREM 3.11.10. Let x (t) e NF . Let 0 (A) be an analytic functions de-
fined on an open set U containing
a(x) _ {z: z = x(t), 0 < t < 27c}. Then the function l(x(t)) also belongs to NF. For N(u) = u we obtain the classical theorem of Levi. For N(u) = uP, 0 < p < 1, Theorem 3.11.10 was proved by 2elazko (1960). Let N satisfy all the conditions described in Example 3.11.7. By NH we denote the space of all analytic functions x(z) defined on the open unit disc D such that the coefficients {xn}, n = 0, 1, ... of the power expansion co
x(z) = f xnzn n=o
form a sequence {xn} belonging to N(1). There is a one-to-one corespond-
ence between pointwise multiplication in NH and the convolution in N(1). Thus we can identify NH with N(l). Now we shall show that every multiplicative linear functional F de-
fined on NH is of the form F(x) = x(zo), Izol < 1. To begin with we shall show this for x(z) = z. Suppose that F(z) = a, Ial > 1. Then F(z/a) = 1. Therefore F(a-nzn) = 1. On the other hand, a-nzn-* 0, and this leads to a contradiction with the continuity of F.
Observe that N(l) C 1. Therefore each function x(z) a NH can be extended to a continuous function defined on the closed unit disc D. Thus we have THEOREM 3.11.11. Let x(z) a NH. Let 0 be an analytic function defined on an open set U containing x(D). Then O(x(z)) a NH.
Locally Pseudoconvex and Locally Bounded Spaces
183
Theorems 3.11.10 and 3.11.11 can be extended to the case of many variables in the following way THEOREM 3.11.12 (Gramsch, 1967; Przeworska-Rolewicz and Rolewicz,
1966). Let x1, ..., xn e N. (or NH). Let 0(z1, ..., zn) be an analytic function of n variables defined on an open set U containing the set
a(x) _ {(xl(t), ...,xn(t)): 0 < t < 2n} a(x) _ {xl(z), ..., xn(z)): I z I < 1}). Then the function 1(x1, ..., x,) belongs to N. (or respectively, to NH).
We shall not give here an exact proof. The idea is the following. Replacing the Cauchy integral formula by the Weyl integral formula, we can define analytic functions of many variables on complete locally bounded algebras.
3.12. LAW OF LARGE NUMBERS IN LOCALLY BOUNDED SPACES
Let (Q, X, P) be a probability space. Let (X, 11 11) be a locally bounded space. Let the norm 11 11 be p-homogeneous. As in the scalar case,
a measurable function X(t) with values in X will be called a random variable. We say' that two random variables X (t), Y(t) are identically distributed if, for any open set A C X
P({t: X(t) e A}) = P({t: Y(t) E A}).
A random variable X(t) is called symmetric if X(t) and -X(t) are identically distributed. Random variables Xl(t ), ..., Xn(t) are called independent if, for arbitrary open sets Al, ..., An
P({t: X{(t) a At, i = 1, ..., n}) _
P({t: XX(t) e At}).
A sequence of random variables {XX(t)} is called a sequence of independent random variables if, for each system of indices n1, ..., nx, the random variables Xnl(t), ..., Xnt(t) are independent.
Chapter 3
184
THEOREM 3.12.1 (Sundaresan and Woyczynski, 1980). Let X be a locally bounded space. Let II II be a p-homogeneous norm determining the topology in X. Let {XX(t)} be a sequence of independent, symmetric, identically distributed random variables. Then
E(IIXIII) = f IIX1(t)IIdP < +oo
(3.12.1)
a
if and only if
X1(t)+ ... + Xn(t) n1IP
0
(3.12.2)
almost everywhere.
Proof. Necessity. To begin with we shall show it under an additional hypothesis that X1 takes only a countable number of values x1, x2, .. . Since X. are identically distributed, all X. admit values x1, ... For each positive integer m we shall define new random variables X k(t)
Xk(t)
if Xk(t) = x1i ..., xm,
0
elsewhere.
By Rk(t) we shall denote Xk(t)-Xk(t). For each fixed m, {IIXk II} constitutes a sequence of independent identically distributed symmetric random variables taking real values. Moreover, E(IIXi II) < E(IIXIII) < +oo.
(3.12.3)
The random variables {Xk } takes values in a finite-dimensional space. Thus we can use the strong law of large numbers for the one-dimensional case (see for example Petrov, 1975, Theorem IX.3.17). Let an = n11P. Then
ak-2 = nL=Jk
n-2IP nL=Jk
= 0(n-2P+I) = O(n an 2).
(3.12.4)
Having (3.12.3) and (3.12.4), we can use the strong law of large numbers
by coordinates (here we use the fact that Xx takes values in a finite dimensional space).
Locally Pseudoconvex and Locally Bounded Spaces
185
Thus
n-11P(Xi + ...
(3.12.5)
-->O
almost everywhere. At the same time, IIRn II is a sequence of independent identically
distributed real random variables with finite expectation, so that, by the classical strong law of large numbers, IIRi II+ ... +II Rn II -*E (IIR1 II) n
(3.12.6)
almost everywhere. Since IIRI II tends pointwise to 0 as m tends to infinity
and IIRmJI < IIX1II, by the Lebesgue dominated convergence theorem E(IIRi II) tends to 0.
The set Slo of those t for which (3.12.5) and (3.12.6) converge at t is of full measure, i.e. P(Q0) = P(Q) = 1 Let e > 0. Choose an m > 0 such that E(IIR II) <
4
For any t e Do, we can find an N = N(e, t, m) such that IIn-IIP(X1(t)+ ... X'(t))II <
(3.12.6)
for n > N. Thus, for t e £2 and n > N, IIn-1/P(X1(t)+ ... +X2(t))II
IIn-11P(Xi (t)+ ...
+Xn(t))II+IIn-1IP(R(t)+
... +R(t))
2 +E(IIRi II) < e. This completes the proof under the condition that X. are countable valued. To complete the proof of necessity we shall use the standard approxi-
mation procedure. For each e > 0, there is a symmetric Borel function T. taking values in a countable set in X such that II TT(x)-xII < e.
(3.12.7)
Chapter 3
186
Hence
Iin-IlP(Xl+... +Xn)II <
*

for x e X.
and (4.1.4)
Proof. We say that a linear functional h(x) is an extension of a linear functional g(x) if the domain of the functional h(x) contains the domain of the functional g(x) and both functionals coincide on the domain of the functional g(x). Let 9T be the set of all extensions of the functional fo such that
g(x) < p(x)
for x belonging to the domain of g.
We can introduce a relation of partial order in 91 in the following way. We say that h --S g if g is an extension of h. By the Kuratowski-Zorn lemma there is a maximal element of the family 91. Let (f(x) be such a maximal element. To complete the proof it is sufficient to show that the domain of the functionalf(x) is the whole space X.
Chapter 4
190
Suppose that the above does not hold, i.e. that the domain Y of the functionalf(x) is different from the whole space X. Let yo e Y and let us denote by Yo the space spanned by Y and yo.
Obviously each element x e Yo can be written uniquely in the form x = y+ayo, where y e Y and a is a scalar. Let x', x" a Y. Then
f(x')+f(x") =f(X +x") Ilfoll
Let us now consider the complex case. We shall define two linear real valued functionals11 and f2 on Y by the equality
fo(x) =fi(x)+if2(x) Of course, IIf ll < Ilfoll and IIf2II < Ilfoll. Moreover, for x e Y,
f1(iX)+if2(iX) =fo(iX) = ifo(X) = ifi(x)-f2(x). ,(ix). Therefore fo(x) = fi(x)-if,(ix). Hence f2(x) = Let Fi(x) be a real-valued norm preserving extension fl(x) to the whole space X. Of course, by definition, IIFIII = Ilfill Letf(x) = Fi(x)-iF,(ix).
The functional f(x) is a continuous functional linear with respect to complex numbers (compare Corollary 4.1.3). Since F, is an extension off,, f is an extension of the functional fo. To complete the proof it is enough to show that I f II < Ilfoll
Let x be an arbitrary element of X. Let O = argf(x). Then If(x)I = If(e-iex)I = IF1(e-i1x)I < IIFiiflleiexII < Ilfi II IIxII < IIfthI IIxII Hence IIxII < Ilfoll.
COROLLARY 4.1.6. Let X be a normed space. Then
IIxII = sup
If(X) I
111111
where the supremum is taken over all continuous linear functionals f e X*. Proof. Let x0 be an arbitrary element of X. Let Xo be the space spanned
by x0, i.e. the space of all elements of the type axo. Let us put fo(x) = a I IxoII for x e Xo. The functional fo is of norm one. Basing ourselves on Theorem 4.1.5, we can extend it to a continuous linear function Fo of norm one. Then II xoll = fo(xo) = Fo(xo). Hence
sup I f(xo) I < I Ixol I = Fo(xo) 111111
< IUII41 sup f(xo)
Existence of Continuous Linear Functionals and Operators
193
Duren, Romberg and Shields (1969) gave an example of an F-space
X with a total family of continuous linear functionals such that X possesses a subspace N such that the quotient space X/N has trivial dual. Shapiro (1969) showed that 12', 0 < p < 1, also contain such subspaces N. Kalton (1978) showed that every separable non-locally convex F-space has this property. 4.2. EXISTENCE AND NON-EXISTENCE OF CONTINUOUS LINEAR FUNCTIONALS
THEOREM 4.2.1 (Rolewicz, 1959). Let
lim inf N(t) > 0. t t->ao Then in the space N(L(Q,E,u)) there are non-trivial continuous linear functionals. Proof. Corollaries 4.1.2 and 4.1.3 imply that it is enough to show that. in the space N(L(Q,E,p)) there is an open convex set U different from
the whole space. Let E e E, where µ(E) = a, 0 < a <+oo. Since lim inf N(t) > 0, there are a positive constant a and a positive number t-aoo
t
T such that, for t > T, N(t) > at. Let U be a convex hull of the set {x: pN(x) < 11. The set U is an open convex set. We shall show that it is different from the whole space N(L(S2,E,µ)). Let x1, ... , x.,, be arbitrary elements such that pN(xt) < 1, i = 1, 2, ..., n
Let Bs = {s: Ixs(s)l > T}. Let xi(s)
for s e BB,
elsewhere
I0
Let X0, =
X1'+ ... +x;,
o
n
and
and
x;' = xs-x;.
... xo" = xi + n
Since Ixa'(s)I < T(i = 1,2,..., n), Ix"(s)I *
not belong to the set U.
If the measure p has an atom E0 of finite measure, then there is a non-trivial continuous linear functional in the space N(L(Q,E,p). Indeed, by the definition of measurable function, x(t) is constant on E0 p-almost everywhere, x(t) = c. Let us put f(x) = c. It is obvious that f(x) is a continuous linear functional.
If the measure p is atomless, the following theorem, converse to Theorem 4.2.1, holds : THEOREM 4.2.2 (Rolewicz, 1959). Let p be an atomless measure. If
lim inf N(t) = 0,
(4.2.1)
t
t- oo
then there are no non-trivial continuous linear functionals in the space
N(L(Q,E,p)) Proof. Let e be an arbitrary positive number. Suppose that (4.2.1) holds.
Then there is a sequence {tm} tending to infinity such that
Ntmm) -->O.
Let E be an
Let km be the smallest integer greater that
arbitrary set belonging to E of the finite measure. Since the measure p is atomless, there are measurable disjoint sets El, ..., Eke, such that km
E= U EE
and
,u(Ei) = (E)
(i = 1, 2, ..., km).
Let xm(s) =
{
tm
for s e EE (i = 1, 2, ..., km),
0
elsewhere.
Obviously, for sufficiently large m, pN(x;,) < e. On the other hand, for s e E km
ym(s) = km
x`m(s) = km Lam, ti=1
Existence of Continuous Linear Functionals and Operators
195
Since km moo, every function of the type aXE belongs to the convex hull U of the set {x: pr,(x) < e}. The set E is an arbitrary set of finite measure. Hence all simple functions with suports of finite measures belong to U. The set U is open and convex. Therefore U = N(L(Q,E,p)).
COROLLARY 4.2.3 (Day, 1940). In the spaces LP(Q,E,u), 0 *

*
Let X be an F-space. We say that a family X' of continuous linear functional is total if f(x) = 0 for all f e X' implies that x = 0. Theorem 4.1.4 implies that, if X is locally convex, then the family of all continuous linear functionals is total.
Let Y be a subspace of the space X. If there is a total family X' of continuous linear functionals on X, then the family X' is a total family of functionals on the space Y. Problem 4.2.4. Does every infinite-dimensional F*-space X contain an infinite dimensional subspace Y with a total family of continuous linear functionals? We do not kpow the solution of the even simpler
Problem 4.2.5. Does every infinite-dimensional F*-space X contain an infinite-dimensional subspace Y with a non-trivial continuous linear functional?
We do not know whether Problems 4.2.4 and 4.2.5 are equivalent. There is also a sequence of weaker questions, for example : Problem 4.2.6. Does every infinite-dimensional locally bounded space
contain an infinite-dimensional space with a non-trivial continuous linear functional? We know only the following sufficient condition.
Chapter 4
196
We say that an F-space Y contains arbitrarily short lines if, for each positive i, there is an element x e Y, x 0, such that sup IItxII < E. t-scalars
PROPOSITION 4.2.7 (Bessaga, Pelczynski and Rolewicz, 1957). If an F-space X contains arbitrarily short lines, then it contains a subspace X0 isomorphic to the space (s) of all sequences (see Example 3.1.a"). Proof. Let r(x) = sup IItxII. Let {e,} be a sequence of elements of X 1-scalars
such that
r(en+l) < 4 r(en).
(4.2.2)
Formula (4.2.2) implies that for each sequence of scalars {tn} the 00
to en is convergent.
series n=1
oo
We shall say that
to en = 0 implies to = 0 for n = 1, 2, .. . n=1
Indeed, suppose that the above does not hold. Let ti be the first scalar different from 0. From the definition of r(x) it follows that there is a number a such that (4.2.3.)
Ilatieill > a r(ei).
On the other hand,
00
at,, en = 0. Hence
n=1
r(en) <
Ilateill < I Ilatnenll <
n=i+1
n=i+1
I4nr(et) n=i+1
/
= a r(ei),
and this contradicts (4.2.2). By a simple calculation we find that if Go
GO
t; ei} is convergent to x =
a sequence {xn} i=1
tte{, then t; -->ti, i=1
n = 1,2, ... and conversely. Therefore, the set of all elements of type 00
E tnen constitutes a subspace Xo isomorphic to (s).
n=1
Existence of Continuous Linear Functionals and Operators
197
COROLLARY 4.2.8. If an F-space X contains arbitrarily short lines, then X contains an infinite-dimensional subspace X° with a total family of continuous linear functionals.
Proof. Proposition 4.2.7 implies that the space X contains a subspace isomorphic to (s) and in the space (s) there is a total family of continuous linear functionals, because the space (s) is locally convex.
Kalton (1979) has shown that any strictly galbed space that is not locally bounded contains an infinite dimensional locally convex space. Let us observe that there is an F-space X with a total family of continuous linear functionals and a subspace Y of the space X such that
in the quotient space X/Y there are no non-trivial continuous linear functionals. Indeed, let X =1P, 0 < p < 1. By Theorem 3:2.12 there is a continuous linear operator T mapping X onto LP[O, 1]. Let Y = T-1(0). Then the quotient space Z = X/Y is isomorphic to LP[0,1], and this implies (see Corollary 4.2.3) that there are no non-trivial continuous linear functionals in Z (see Shapiro, 1969). Klee (1956) has shown that there is a separable F*-space X with a total family of continuous linear functionals such that for each integer n > 2 there is a system of subspaces M1, ... , M of X such that M2 n M3
= {0} for i #j, M1+...+M = X, and for each i the quotient space X/Mt does not admit non-trivial continuous linear functionals for any i. Let X be an F-space. If there are no non-trivial linear continuous func-
tionals defined on X, then X is not isomorphic to its product by the one-dimensional space X x R (X x C), since in this product there is a non-trivial linear continuous functional. A more complicated problem is posed by quotient spaces.
Problem 4.2.9. Suppose that in an F-space X there are no non-trivial linear continuous functionals. Is X not isomorphic to the quotient of X by a one-dimensional space?
A partial answer to the question was given by Kalton and Peck, (1979). They showed that if B1 and B2 are two locally bounded subspaces of L°[0,1], then the quotient spaces L°[0,1]/B1 and LO[O, 1]/B2 are
isomorphic if and only if there is a continuous one-to-one operator T mapping L°[0,1] onto itself such that T(B1) = T(B2). In particular
Chapter 4
198
L°[0,1]/En is isomorphic to L°[0,1]/Em, where Ej, i = 0, 1, 2, ..., denote an i-dimensional subspace of L°[0,1] if and only if n = m. Let us consider functions x(t) of the real argument t, 0 < t < 1, with values in an F-space X. In the same way as in the calculus we can define
the derivatives. We say that a function c(t) has a derivative at point t if there is a limit
x'(t) = lim
x(t+h)-x(t) h
In the calculus there is a theorem which states that if a function x(t) has a derivative at each point and that derivative is equal to 0 on the whole interval, then the function x(t) is constant. PROPOSITION 4.2.10. Suppose that in an F-space X there is a total family
X' of continuous linear functionals. Then each function x(t) defined on the interval [0, 1] with values in the space X and such that the derivative x'(t) exists and is equal to 0 at each point t is constant. Proof. Let f e X'. We consider the scalar valued function F(t) = f(x(t)). We have :
F(t) - lim f(x(t+h))-f(x(t)) h h- o
=f
(lim
x(t+hh-x(t)) =f(x'(t))
o
Hence, if x'(t) = 0, then f(x'(t)) is equal to 0 for all f e X'. Therefore, by the above mentioned theorem from the calculus, f(x'(t)) is constant. The totality of the family X' implies that x(t) is constant. Without the assumption of the existence of a total family of continuous
linear functionals the statement is not true, as follows from THEOREM 4.2.11 (Rolewicz, 1959b). Let x0 be an arbitrary element in
the space S[0,1]. Then there is a function x(t) defined on the interval [0, 1] with values in the space S[0,1] such that :
(1) x(0) = 0, (2) x(1) = x0,
(3)x'(t)=0 f o r allt,0 *

*
Example 4.3.1a' Each continuous linear functional defined on the space 1P, 1 < p < +oo, is of the form co
F(x)
Yn xn ,
(4.3.2)
n=1
where {y,,} a 14. The norm of the functional F is given by the formula IIFII = II{yn}II,4
Example 4.3.2
Each continuous linear functional F(x) defined on the space L(Q,E,,u) with the norm IIxil = f Ix(t)I dp is of the form a
F(x) = f y(t)x(t)dµ.
(4.3.3)
a
where y(t) e M(Q,E,p) and the norm of the functional F is given by the formula IIFII = esssuplY(t)I LEO -
(i.e. IIFII = IIYII(a,E,ju))
Existence of Continuous Linear Functionals and Operators
201
Example 4.3.2a Each continuous linear functional F defined on the space 1 is of the form
(4.3.2), where {y,,} a m. The norm of the functional F is given by the formula IFII = max Iynl.
Example 4.3.3
Each continuous linear functional F(x) defined on the space C(Q) is of the form
F(x) = f x(t)du,
(4.3.4)
a
where y is a countable additive scalar valued function defined on Borel subsets of Q. Moreover, It is regular (i.e. for any Borel set E and for any positive a there are an open set G and a closed set F such that FC EC G,
and Ip(G\F)I < e) and the total variation is bounded, V(p, Q) =
sup Ip(E,)I+Ip(E2)I < +00. E,vE,=a
Ei Borel sets
If we take in C(Q) the standard norm IxII = sup Ix(t) I tea
then the norm of the functional F is equal to the total variation of the measure. Example 4.3.3a In the space C[O,1] each continuous linear functional is of the form
i F(x) = f x(t)dy(t),
(4.3.5)
0
where y(t) is a function of the bounded variation. The norm of the functional F is given by the formula IFII = Var y(t). 0
Chapter 4
202
Example 4.3.3b In the space c each continuous linear functional is of the form co
F(x) = Lam, ynxn+Yo lim xn, n=1
(4.3.6)
n->co
where {yp} e 1 and the norm of the functional F is given by the formula W
IFI1=IYoI+fIYnI n=1
Example 4.3.3b' Each continuous linear functional defined on the space co is of the form (4.3.6), where of course yo = 0. Example 4.3.4
In the Hilbert space H each continuous linear functional is of the form F(x) = (x, y), (4.3.7) and the norm of the functional F is equal to the norm of y. Formulae (4.3.2)-(4.3.7) define continuous linear functionals in the respective spaces. The proof of formulae (4.3.2)-(4.3.7) can be found for example, in Dunford and Schwartz (1958). The proof of formula (4.3.1) is given in Krasnosielski and Ruticki (1958). 4.4. CONTINUOUS LINEAR F UNTIONALS IN Bo SPACES
Let X be a Bo-space with the topology determined by a sequence of homogeneous pseudonorms {IIxjIn}. As a simple consequence of the considerations of Section 4.1 we obtain THEOREM 4.4.1 (Mazur and Orlicz, 1948). Let X be a B, -space with the
topology determined by a sequence of pseudonorms A linear functional f(x) defined on the space X is continuous if and only if there {Ilxll,,}.
are a pseudonorm I IxJI., and a constant Kf such that
I *f(x) I < KiIxjI.,
Existence of Continuous Linear Functionals and Operators
203
COROLLARY 4.4.2. Let X be a Bo-space with the topology determined by a sequence of pseudonorms IIxIIn. Let
X° = {x: IIxIIn = 0}
and
XU = X/Xn.
The pseudonorm IIxIIn induces the norm in the space X. Then the con-
jugate space X* is the union of the space X*
= U Xn. n=1
Let us remark that if IIXIII , IIXII2 < IIXII3 < ...,
then Xn C X;,+1
and
IIfII1 >
IIf1I2 >
IIfII3> ...
for norms of functionals. The topology of bounded convergence in X is equivalent to the topology given by the following basis of neighbourhoods of zero: U = f f. fE Xk, I If I Ik < s for a positive s and an index k}.
If X is not a B*-space, then the space X is not metrizable; in fact, if X is not a B*-space, then there is a system of pseudonorms IIxII1 < I Ix I I2 < ... determining the topology equivalent to the original one such that X, Xn_1.
Let fn be an arbitrary sequence of functionals such that fn E Xn and fn 0 X.'- 1.
Of course, for any sequence of scalars {tn}, to tn,fnEXn
and
0, n = 1, 2, ...
tnfn0Xn_1.
Let U be an arbitrary neighbourhood of zero in X *. It is easy to verify that only a finite number of elements tafn belong to U. Since {tn} is an arbitrary sequence of scalars different from 0, the space X* is not
a linear metric space. Theorem 4.4.1 and the knowledge of the general form of continuous linear functionals in Banach spaces permit us to give the general form of continuous linear functionals in B0-spaces
Chapter 4
204
Example 4.4.3
Each continuous linear functional defined on the space e0(Q) (see Example 1.3.6) is of the form
F(x) = f x(t)dp,
(4.4.1)
nx
for a certain k and some measure p satisfying the conditions described in Example 4.3.3. Example 4.4.4
Each continuous linear functional defined on the space C°°(91) (see Example 1.3.7) is of the form
f F(x) =
J atk=a..3tk^ x(t)dpk
(4.4.2)
Ikl<m 92
for a certain positive integer m and some measure µk satisfying the conditions described in Example 4.3.3. Example 4.4.5
Each continuous linear functional on the space LP(am, k), 1 < p <+00, is of the form
F(x) _ f Yn xn
(4.4.3)
n=1
where {yn} is such a sequence that, for a certain index m, CO
Y
yn
n= 1
Iq
1
am n
<+
C'O
and 1/p+l/q = 1. Example 4.4.6
Each continuous linear functional defined on the space L(am,n) is of the form (4.4.3), where {yn} is such a sequence that, for a certain index m,
supyn
1
am,n
I<+00.
Existence of Continuous Linear Functionals and Operators
205
THEOREM 4.4.7 (Eidelheit, 1936). Let X be a Bo-space. Let {fn} be a sequence of linearly independent continuous linear functionals defined on X. Let us suppose that for any continuous pseudonorm IIxII there is only a finite number of functionals fn,, ...,fn, continuous with respect to the pseudonorm IIxII (i.e. for i nk sup { f (x) : IIxII < 1} = +oo). Then for any sequence of scalars {an} there is in X such an element x0 that fn(xo) = an.
Proof. Let IIxIIk be an increasing sequence of pseudonorms determining the topology IIxII1*

Lq such that TI = f (here 1 denotes a constant function equal to one) and [Tx]q < [ f]q [x]p for x e LP. (4.6.9) Proof. To begin with we shall prove the theorem for p = q. Suppose that inflf(t)I > 0. Let f 0 f Ulq Thus [fo]q = 1. Let T0(x) = x(F(t))fo(t ), where t F(t) = f Ifo(t) qdt. 0 The operator TO is linear and I X(F(t))Iq dF = IIXIIq 1T'0(X)IIq = f I x(F(t))fo(t)j" dt = 0 0 It is easy to verify that TI = [f]gT01 = f(t). We recall that, in general, Lq D LP and, for y e LP, [y]q < [ylp Thus [Tx]q < [X]q[f ]q < [X]p[f]q. (4.6.9) holds provided inf I f(t)I > 0. Since the functions f with this property are dense in Lq, by continuity arguments we find that (4.6.9) holds for all functions belonging to Lq. Existence of Continuous Linear Functionals and Operators 213 Let C be a one-dimensional subspace of the space LP, 0

*LP/C be the quotient map associating with each x E LP the coset containing x. Of course by the definition of the quotient space IIp0fIl, = inf {Ilf--allP: a being scalar} where II II denotes the quotient norm. Since there is no danger of confusion, we shall denote II Ii; also by II IIP
The space LP/c, 0 < p < 1 is not isomorphic to the space LP (see Kalton and Peck, 1979); nevertheless, it embeds into LP, as follows from
LEMMA 4.6.3. There is a linear operator S : LP-*LP, 0 < p < 1, such that IIpofIIP < IIS(f)IIP < 2IIpoflIP
(4.6.11)
Proof. By the classical results of measure theory the space LP is isometric to the space LP([0,1] x [0, 1]). We define S : LP->.LP([0,1] x [0, 1]) by Sfl(x.y)
=f(x)-f(y)
Then
ii
i
IISf1IP = f f If(x)-fly) IPdxdy> f IlpofIIPdy = Ilpofllp 0
0
0
On the other hand, IISfIIP I *

*
Chapter 5
Weak Topologies
5.1. CONVEX SETS AND LOCALLY CONVEX TOPOLOGICAL SPACES
Let X be a linear space over real or complex numbers. A set A C X is said to be convex if, for arbitrary non-negative, a, b such that a+b = 1, x, y e A implies ax+by c- A (see Section 3.1). The intersection of an arbitrary family of convex sets is a convex set. Let A be a convex set and let a1, ..., an be non-negative numbers such
that
at= 1. i=1
Then n 1
ajxt e A
i=1
for an arbitrary system of elements x1, ..., xn c- A. Let A be an arbitrary set. By conv(A) we denote the intersection of all convex sets containing A. The set conv(A) is called the convex hull
of the set A. It is easy to verify that the convex hull of the set A may be characterized in the following way : in
in
anxn: xn E A, an > 0, 1 an = 1 j.
conv(A) n=1
(5.1.1)
n=1
The algebraic sum of two convex sets is a convex set. The image of a convex set A under a linear operator T is a convex set. A counterimage of a convex set under a linear operator is a convex set.
Let M be an arbitrary subset of X. We say that a point p c- M is 221
Chapter 5
222
a C-internal point of the set M if, for each x e X, there is a positive number a such that p+tx e M for It I < e. A point p e X is called a C-bounding point of the set M if it is neither a C-internal point of the set M nor a C-internal point of the complement of M.
Let K be a convex set containing 0 as a C-internal point. Then we can define a functional p (x) (generally non-linear) in the following way :
p(x)=inf{t>0: t eK}.
(5.1.2)
Let us remark that the functional p(x) has the following properties (compare Section 4.1) : (1) p (x) > 0,
(2) p(x) <+oo for x e lin K (3) p(x) = tp(x) for positive t, (4) p (x+y) < p (x) +p (y), (5) if xe K, then p(x) 1, (6) if x is a C-internal point of the set K, then p (x) < 1, (7) if x is a C-bounding point of K, then p (x) = 1.
We say that a linear functional f(x) separates two sets M and N if there is a constant c such that Re f(x) > c for x e M and
Re f(x) < c
for x e N. Here by Re z we denote the real part of a complex number z. Obviously, if X is a linear space over reals, then Re f(x) = f(x). Of course, a functional f(x) separates the sets M and N if and only if it separates the sets M - N and {0}. PROPOSITION 5.1.1. Let M and N be two disjoint convex sets in a linear
space X. Suppose that M has a C-internal point. Then there is a linear functionalf(x) separating the sets M and N. Proof. Without loss of generality we can assume that 0 is a C-internal point of the set M. To begin with, we shall consider the case where X is a linear space
over reals. Let -y be an arbitrary C-internal point of the set M-N.
Weak Topologies in Banach Spaces
223
Since the sets M and N are disjoint, the point 0 does not belong to the
set M-N, Let K = M-N+y Obviously, 0 is a C-internal point of the set K and the point +y does not belong to K. Let p(x) be the functional defined by formula (5.1.2). Obviously, p(+y) > 1. Let X0 be the space spanned by the element y. We define on the space X0 a linear functionalfo in the following way fo(ay) = ap(y). Of course, for x e Xo, .fo(x) < P (x)
By the Hahn-Banach theorem (Theorem 4.1.1) the functional fo(x) can be extended to the functional f(x) defined on the whole space X such thatf(x) < p (x). This implies thatf(x) < 1 for x e K and fly) > 1. Hence the functional f separates the sets M-N and {0}. Therefore, it separates the sets M and N. Now we shall consider the complex case, i.e. the case where X is a linear space over complex numbers. The space X may obviously be considered as a linear space over reals, too. Then there is a real-valued
linear functional f(x) separating the sets M and N. Let g(x) = f(x) -if(ix) (cf. Corollary 4.1.3). The functional g(x) is linear with respect to complex numbers and it separates the sets M and N. We recall that X is a linear topological space if it is a linear space with
a topology and if the operations of addition and multiplication by scalars are continuous.
Let A be a subset of a linear topological space X, If A is a convex set, then the closure A of the set A is also a convex set and the interior Int(A) of the set A is also convex. The continuity of multiplication by scalars implies that each internal point of the set A is a C-internal point of this set. If a convex set A contains internal points, then the necessary
and sufficient condition for the point x e A to be C-internal is that x be an internal point. By conv(A) we denote the intersection of all closed convex sets containing A. The set conv (A) is called the closed convex hull of the set A. It is easy to verify that (5.1.3) conv (A) = conv(A), (5.1.4) conv (aA) = a conv (A) for all scalars a.
Chapter 5
224
PROPOSITION 5.1.2. If cony (A) is a compact set, then
cony (A +B) = cony (A)+ conv (B).
The above formula is a consequence of the following lemma : LEMMA 5.1.3 (Leray, 1950). Suppose we are given a topological space Y, a compact space K, and a continuous mapping f(t, k) of the product X x K into a topological space X. Let F be a closed subset of the space X disjoint with the set f(to, K). Then there is a neighbourhood V of the point to such that the set F does not have common points with the set f(V,K).
Proof. Let k e K. The continuity of the function f implies that there are a neighbourhood V(k) of the point to and a neighbourhood W(k) of the point k such that Fnf(V(k), W(k)) = ¢. The set K is compact, hence we can choose a finite cover of the set K by the sets W(kl), ..., m
W(km), Kc U W(ki). Let z=i
m
n V(k{). v= %=I The set V is an open set with the required properties. LEMMA 5.1.4 (Leray, 1950). Let X be a linear topological space. Let F be
a closed set in X and let K be a compact set in X. Then the set F+K is closed.
Proof. If x 0 F+K, then the set F does not have common points with the set x-K. Therefore, by Lemma 5.1.3, there is a neighbourhood V of the point x such that Fn (V-K) Hence
vn(F+K) =-0.
El
The proof of Proposition 5.1.2 is a trivial consequence of Lemma 5.1.4 and of the fact that the algebraic sum of two convex sets is convex.
Let us note some further properties of convex sets. By similar considerations to those in Corollary 4.1.2 we can prove that if a functional f(x) separates two sets M and N and one of them contains an internal point, then the functional f(x) is continuous.
Weak Topologies in Banach Spaces
225
Conversely, Proposition 5.1.1 implies that if M and N are two disjoint
convex sets and one of them contains an internal point, then the sets M and N can be separated by a continuous linear functional f(x). We say that a linear topological space is locally convex if each neighbourhood of zero contains a convex neighbourhood of zero (compare Section 3.1).
THEOREM 5.1.5. Let X be a locally convex topological space. Let M and N be two disjoint convex closed sets. If, in addition, M is compact, then
there are constant c, and e > 0 and a continuous linear functional f(x) defined on the space X such that
Ref(x) < c-e
for xe N,
Ref (x) > c
for x e M.
(5.1.6)
Proof. Lemma 5.1.4 implies that the set M-N is closed. Since M and N are disjoint, the set M-N does not contain the point 0. The space X is locally convex, and therefore there is a convex balanced neighbourhood
of zero U such that Un (M-N) = 0. Proposition 5.1.1 implies that there is a continuous linear functional f(x) which separates the sets U and M-N. The set U is open, whence
sup{Ref(x) : x e U} = e > 0. Therefore, Ref(x)
(5.1.7)
e for x e M-N. This trivially implies the prop-
osition. COROLLARY 5.1.6. Suppose that in a linear space X there are two locally convex topologies (X, Tl) and (X, a2). If both topologies designate the same
continuous linear functionals, then a convex set A is closed in the first topology Tl if and only if it is closed in the second topology T2. Proof Let K be a convex set closed in the topology r1. Then for each point p 0 K there is a linear functional f(x) which is continuous in the
topology Tl and there are numbers c,e > 0, such that Ref(x) < c-e for x e K and Ref(p) > c. But the hypothesis implies that the functional f(x) is continuous in the topology T2. Hence p could not belong to the closure K2 of the set K in the topology T2. This implies that K2 = K.
Chapter 5
226
If K is a convex set closed in the topology r2, then using the same arguments we infer that it is closed in the topology r l.
5.2. WEAK TOPOLOGIES. BASIC PROPERTIES
Let X be a linear space over real or complex numbers. Let X' denote the set of all linear functionals defined on the space X. A subset T of
the set X' is called total if f(x) = 0 for all f e T implies that x = 0 (compare Section 4.2).
Let T be a total linear set of functionals. By the f-topology of the space X we shall mean a topology determined by the neighbourhoods of the type N(p,.fi,... , fx ; ar, ... , ax) = {x: l fc(x-p)I < a; (i = 1, 2,
..., k)},
where at > 0 and f¢ e T. Obviously, the space X with a f-topology is a locally convex space. We say that a set A C X is f-closed (f-compact) if it is closed (compact) in the f-topology. The closure of a set A in the f-topology will be called the f-closure. A functional f(x) is said to be T-continuous if it is continuous in the I'-topology. Let X be a locally convex topological space. Let X* be the set of all
continuous linear functionals defined on X (conjugate space). The X*-topology is called the weak topology. Obviously, the weak topology is not stronger than the original one. Let X be the space conjugate to a locally convex space X_. Let us recall that in the conjugate space we have the topology of bounded convergence. Each element x_ E X_ induces a continuous linear functional F on the space X by the formula
F(x) = x(x_).
(5.2.1)
We shall indentify the set of functionals defined by formula (5.2.1) with the space X_. The X_-topology in X is called the weak topology of
functionals or the weak-*-topology. Since we always have X* ) X_, the weak topology of functionals is not stronger than the weak topology.
Weak Topologies in Banach Spaces
227
PROPOSITION 5.2.1. Let X be a linear space. Let T be a total linear set of functionals. A linear functional f(x) is continuous in the T-topology if and only if f e F.
The proof is based on the following lemma : LEMMA 5.2.2. Let X be a linear space. Let g, fl, ... , fn be linear functionals defined on X. If
fi(x)=0,
i= 1,2,...,n,
implies
g(x) = 0, then g(x) is a linear combination of the functionals fl, ... , fn.
Proof. Without loss of generality we can assume that the functionals ... , f, are linearly independent. Let
Xo={xeX: f{(x)=0,i=1,2,...,n}.
(5.2.2)
Let X be the quotient space X/Xo. The functionals fl, ..., f, induce linearly independent functionals f1, ..., fn on X. The assumption about g(x) implies that the functional j (x) also induces a linear functional (x) defined on X.
Then space X is n-dimensional, therefore, j (x) is a linear combination off , ..., fz,
g=aifl+...+anfn. It is easy to verify that g = aifi+ ... +anf . Proof of Proposition 5.2.1. Sufficiency. From the definition of neighbourhoods in the f-topology it trivially follows that each functional f E f is f-continuous (i.e. continuous in the f-topology). Necessity. Let g(x) :f- 0 be a functional continuous in the f-topology. Then there is a neighbourhood of zero U in that topology such that sup Ig(x)I < 1. But the neighbourhood U is of the type
U={xeX: If(x)I *

(5.3.8)
Hence x** = n(xo). This means that w*(n(A)) C n(X). Since n is a homeomorphism between X and n(X), both with X *-topology, the weak closure of the set A is exactly w*(n(A)). The Alaoglu theorem (Theorem 5.2.4) implies that the set w*(n(A)) is compact.
5.4. EXAMPLE OF AN INFINITE-DIMENSIONAL BANACH SPACE WHICH IS NOT ISOMORPHIC TO ITS SQUARE
In the majority of known examples of infinite-dimensional Banach spaces, those spaces are isomorphic to their Cartesian squares. Now we shall give an example which shows that this is not true in general. The example is based on the following LEMMA 5.4.1 (James, 1951). Let X be Banach space with a basis {Xn}. Let X. denote the space spanned on the elements en+l, en+2, ... If for any functional f belonging to X*
lim if Ix 11 = 0,
(5.4.1)
n->co
where f l y denotes the restriction of the functional f to a subspace Y, then the basis functionals {fn} (see Corollary 2.5.3) constitute a basis in X*.
Weak Topologies in Banach Spaces
235
Proof. Let f e X*. Let x e X, x = 2, fa(z)e.. Then n=1
m
O
0
f(x) = ffn(x)en) = .fn(x)f(en) = n=1
n=1
f L
f n=1
l f(en)f.](x).
00
f(en)fn is convergent to f in
Formula (5.4.1) implies that the series n=1
the norm of the functionals and that this expansion is unique. Example 5.4.2 (James, 1951)
Let x = {x1,x2, ...} be a sequence of real numbers. Let us write n
IIxII = sup
(xp=i_1-xp,t)2+(xp2n+1)2]1/2,
ti=1
where the supremum is taken over all positive integers n and finite increasing sequences of positive integers pl, ..., Pen+1 Let B be a Banach space of all x such that IIxII is finite and
lim xn = 0.
n-*,o
IIxII is a norm. Indeed, IIxII = 0 if and only if x = 0, IItxjl = It! IIxII for all scalars t. Now we shall show the triangle inequality. Let x = {x1, x2, ... }, y = {y1, y2, ... }. From the definition of the norm IIx+yiI it follows that for any positive a there is an increasing sequence of indices p1, ... , P2n+1 such that n
IIx+YII <
(xpzi_1+Yp2i-1-xpz,-Ypzi)2+(xpzn+l+ ti=1
71/2+e
+Ypzn+1)2
n 1/2
(xp2i-l-xps1)2+(xpzn+l)2,
[ i=1
(ypai-l-Yp9i)2+(Ypzn+1)211/2+E
+
ti=1
Thus the arbitrariness of a implies that IIx+YII < IIxII+IIYII
Let zn = {0, ..., 0, 1, 0, ...}. n-th place
< IIxII+IIYII+e
Chapter 5
236
It is easy to verify that the linear combinations of the elements zn are dense in the whole space B, because lim xn = 0. Moreover, for all positive integers n and p, n+p
n
aiziJI < 11 V atz= i=1
t=1
Then Theorem 3.2.15 implies that {zn}is a basis in B. We shall show now that the space B is not reflexive. For this purpose
we shall prove that the closed unit ball in B is not weakly compact. Let yn = zl+ ... +zn. Of course, IIy,II = 1. If the closed unit ball is weakly compact, then {y.} converges to a y e B. Since {zn} is a basis, y ought to be of the form (1, 1, ...). This is impossible, because, for all
xeB,limx,=0. co
Now we shall describe all functionals f belonging to the second conjugate space B**. Let {gn} be the basis functionals with respect to the basis {en}. According to Lemma 5.4.1, {gn} constitute a basis in the conjugate space B*. Let F be a functional from B**. Then the functional F is of the following form : there is a sequence of real numbers 00
00
{Fi} such that F(f) = j' Fi fi for any f e B, f = T' f gi. Let us calculate the norm of the functional F. We have n
n
i=1
i=1
= IIf!I sup[
f n
Fif{I =If(f Ftzi) <11fil I
Fizi
i=1
i=1
where the supremum is taken over all positive integers n' and increasing sequences of indices p1, ..., pen'+1 with Fpk replaced by 0 if pk > n. The arbitrariness of n implies n
IIFII < sup [ i=1
(Fptt_1_Fp2t)2+(Fp2n'+1)211/2. (5.4.2)
Weak Topologies in Banach Spaces
237
Let us now fix n and let un = S' Fizz. Let us define a linear functional i=1
f on the space Y. spanned by the elements un, zn+1, zn+2, ... in such a way that f(aun) = IlunIl
f(zi) = 0
and
for i = n+1, n+2,...
Then 00
00
f (aun+ Y a;zi)= Ilaunll < Ilunn+ f aizi i=n+1
Thus IIFII = 1. Let j=
i=n+1
figi be an extension of the functional f to
whole space B of norm one. Then fi = 0 for i > n and n
oo
F(f)=
i=1
Fifi' =I fFifi i=1
= I f(un)I = IlunIl < IIFII
Hence, calculating the norm of un, we obtain O
IIFII i f L
i=1
`Fp, 1-Fp, )2+(FPan+1)2]1/2
(5.4.3)
for all positive integers n and all finite increasing sequences of integers P1, , p2,+1 Combining (5.4.2) and (5.4.3), we obtain IIFII = sup[
i=1
(5.4.4)
(FP,i_1_FP:J2+(FPan+)2]1/2,
where the supremum is taken over all positive integers n and finite increasing sequences p1, ..., p2,+1. The norm IIFII is finite if and only if
there is a limit lim F. Since the space B is not reflexive, B** contains n=oo
an element which does not belong to n(B). Then the only possibility is B**
that n (B) is a subspace of codimension 1, i.e. that dim n(B) = 1. There(BXB)** X**
fore, dim n(B X B) = 2, and since dim is an invariant of an T, -(X-) isomorphism, the space B is not isomorphic to its Cartesian square (see Bessaga and Pelczynski, 1960b).
Chapter 5
238
Pelczyliski and Semadeni (1960) showed another example of a space
which is not isomorphic to its square. Their example is of the type C(SC). An example of a reflexive Banach space non-isomorphic to its square was given by Figiel (1972).
Problem 5.4.3. Does there exist a Banach space non-isomorphic to its Cartesian product by the real line ? The answer is positive for locally convex spaces, as will be shown in Corollary 6.6.12. Rolewicz (1971) gave an example of normed (non-complete) space X non-isomorphic to its product by the real line. Dubinsky (1971) proved
that each Bo-space contains a linear subset X which is not isomorphic to its product by the real line. Bessaga (1981) gave an example of a normed space which is not Lipschitz homeomorphic to its product by the real line. 5.5. EXTREME POINTS
Let X be a linear space over the real or the complex numbers. Let K be an arbitrary subset of X. We say that a point k e K is an extreme point of the set K if there are no two points k,, k2 e K and no real number
a, 0 < a < 1 such that k = ak,-{-(1-a)k2.
(5.5.1)
The set of all extreme points belonging to K is denoted by E(K). A subset A of the set K is called an extreme subset if, for each k e A the existence of k,, k2, 0 < a < 1 satisfying (5.5.1) implies that k,, k2 a A. PROPOSITION 5.5.1. Let X be a locally convex topological space. Let K be a compact set in X. Then the set E(K) is non-void.
Proof. Let 2C be a family of extreme closed subsets of the set K. We can
partially order this family in the following way : we say that A - B if B D A. Since the set K is compact, the intersection of a decreasing family of closed sets is a closed non-void set, and obviously it is also an extreme set, provided the members of the family belong to 2I.
Weak Topologies in Banach Spaces
239
Then, by the Kuratowski-Zorn lemma, there is a minimal element A0 of the family W.
We shall show that the set A0 contains only one point. Indeed, let us suppose that there are two different points p, q e A0. Then there is a functional x* e X* such that Rex*(p) Rex*(q). (5.5.2) Let
A, = {x e Ao: Rex*(x) = inf Rex*(y)}.
(5.5.3)
UEAo
Since the set Ao is compact, the set Al is not empty, Moreover, formula (5.5.2) implies that the set Al is a proper part of the set A0. Let k,, k2 be points of K such that there is an a, 0 < a < 1, such that ak,+(1-a)k2 a Al. (5.5.4)
Since Ao is an extreme subset, k, and k2 belong to A0. Since (5.5.4) a Re x*(k,) + (1 - a) Rex*(k2) = inf Rex*(y). yEAo
This is possible if and only if Rex*(kl) = Rex*(k2) = inf Rex*(y). HEA,
This implies that k,,k2 e Al. Hence A, is an extreme set. Thus we obtain
a contradiction, because A0 is a minimal extreme subset. Therefore, A. is a one-point set, A0 = {x0} and, from the definition, x0 is an extreme point. THEOREM 5.5.2 (Krein and Milman, 1940). Let X be a locally convex topo-
logical space. Let K be a compact set in X. Then
cony E(K) D K. (5.5.5) Proof. Suppose that (5.5.5) does not hold. This means that there is an element k c K such that k 0 conv E(K). Then there are a continuous linear functional x* and a constant c and positive a such that Rex*(k) < c (5.5.6) and
Rex*(x) > c+e
for xe conv E(K).
(5.5.7)
K1= {x e K: Rex*(x) = inf Rex*(y)}.
(5.5.8)
Let yew
Chapter 5
240
Since the set K is compact, the set K1 is not empty. By a similar argument
to that used in the proof of Proposition 5.5.1, we can show that K1 is an extreme set. By formula (5.5.7) the set K1 is disjoint with the set E(K). This leads to a contradiction, because, by Proposition 5.5.1, K1 contains an extremal point. COROLLARY 5.5.3. If a set K is compact, then
cony K = cony E(K), COROLLARY 5.5.4. For every compact convex set K,
K = conv E(K). PROPOSITION 5.5.5. Let X be a locally convex topological space. Let Q be a compact set in X such that the set conv Q is also compact. Then the extreme points of the set conv Q belong to Q. Proof. Let p be an extreme point of the set conv Q. Suppose that p does not belong to the set Q. The set Q is closed. Therefore, there is a neigh-
bourhood of zero U such that the sets p+ U and Q are disjoint. Let V be a convex neighbourhood of zero such that
V-V C U. Then the sets p+ V and Q+ V are disjoint. This implies that p e Q+ V. The family {q+ V: q e Q} is a cover of the set Q. Since the set Q is compact, there exists a finite system of neighbourhoods of type qi+ V, n
i = 1,2, ..., n, covering Q, QC U (qi+V). i=1
Let
Ki = conv ((qi+V) n Q). The sets Ki are compact and convex ; therefore
conv(K1 u ... u Kn) = conv (K1 u ... U Kn) = conv Q. Hence n
patki, at i=1
n
0, i=1
at=1, kiaKi.
Since p is an extreme point of conv Q, all at except one are equal to 0.
Weak Topologies in Banach Spaces
241
This means that there is such an index i that
peKi C Q+V, which leads to a contradiction. REMARK 5.5.6. In the previous considerations the assumption that the space X is locally convex can be replaced by the assumption that there is a total family of linear continuous functionals I' defined on X. Indeed, the identity mapping of X equipped with the original topology into X
equipped with the F-topology is continuous. Thus it maps compact sets onto compact sets. Therefore, considering all the results given before in the space X equipped with the T-topology we obtain the validity of the remark.
5.6. EXISTENCE OF A CONVEX COMPACT SET WITHOUT EXTREME POINTS
Roberts (1976, 1977) constructed an F-space (X,
II
II) and a convex
compact set A C X, such that A does not have extreme points. The fundamental role in the construction of the example play a notion of needle points (Roberts, 1976). Let (X, II). be an F-space. We say that a point x0 e X, x0 0, is a needle point if for each E > 0, there is a finite set FC X such that II
x0 e cony F,
(5.6.1)
sup {MMxjI : x e F} < e,
(5.6.2)
cony {0, F} e cony {0, xo}+B8i
(5.6.3)
where, as usual, we denote by BE the ball of radius e, Be = {x: IIxii < E}.
A point xo is called an approximative needle point if, for each E > 0, there is a finite set F such that (5.6.2) and (5.6.3) hold, and moreover xo a conyF+B8.
(5.6.4)
Since E is arbitrary, it is easy to observe that xo is a needle point if and only if it is an approximative needle point. Let E denote the set of all needle points. The set Eu {0} is closed.
Chapter 5
242
From the definition of needle points and the properties of continuous linear operators we obtain PROPOSITION 5.6.1. Let X, Y be two F-spaces. Let T be a continuous linear operator mapping X into Y. If x0 e X is a needle point and T(x0) # 0, then T(xo) is a needle point.
x0
We say that an F-space (X, II ID is a needle point space if each x0 e X, 0 is a needle point.
The construction of the example is carried out in two steps. In the first step we shall show that in each needle point space there is a convex compact set without extreme points, in the second step we shall show
that a large class of spaces (in particular, spaces LP, 0 < p < 1) are needle point spaces. THEOREM 5.6.2 (Roberts, 1976). Let (X, II ID be a needle point F-space. Then there is a convex compact set E C X without extreme points.
Proof. Without loss of generality we may assume that the norm II II is non-decreasing, i.e. that IItxUI is non-decreasing for t > 0 and all x e X. Let {En} be sequence of positive numbers such that co
fEn < X00.
(5.6.5)
n=o
Let xo # 0 be an arbitrary point of the space X. We write E0 = conv({0,xo}). Since X is a needle point space, there is a finite set F = El = {x', ..., x,} such that (5.6.1)-(5.6.3) holds for e = e0. For each x;, i = 1, ..., n1, we can find a finite set F; such that
x; a conv({0} u F;),
(5.6.6)1
sup {IIxJI : x e Fl} < nl ,
(5.6.7)1
1
conv ({0} u F;) C conv {0, xi }+BB, .
(5.6.8)1
?it
Observe that (5.6.8)1 implies
conv({0} u E2) C conv({0} u EI)+Be,,
(5.6.9)1
Weak Topologies in Banach Spaces
243
where n,
E2=
F. 1
M
(5.6.10)1
The set E2 is finite, and thus we can repeat our construction. Finally, we obtain a family of finite sets E. such that for each x e En we have x e conv ({0} u En+1),
(5.6.6)19
sup {IIxjI : x e En} < sn,
(5.6.7)19
conv({O} u E,,+1) C conv({O} v En)+BE,.
(5.6.9)19
Let OD
Ko = conv (U En u {0}). n=0
The set Ko is compact, since it is closed and, for each s > 0, there is a finite s-net in Ko. Indeed, take no such that Co
En<E. n=n, no
By (5.6.9),, the set U E. constitute an s-net in the set Ko. 19=0
Observe that no x
0 can be an extremal point of Ko, since 0 is the Co
unique point of accumulation of the set U En, and, by construction, n=o
no x e En is an extremal point of Ko. Thus the set KO-Ko does not have extremal points.
Now we shall construct a needle point space. Let N(u) be a positive, concave, increasing function defined on the interval [0,+oo) such that N(0) = 0 and lim N(u) = 0. n-o
u
(in particular, N(u) could be uP, 0 < p < 1).
Let Q _ [0,1]' be a countable product of the interval [0, 1] with the measure µ as the product Lebesgue measure. Let E be a o-algebra induced
Chapter 5
244
by the Lebesgue mesurable sets in the interval by the process of taking product. 1
Take now any function f(t) e L°°[0,1] such that f f(t)dt = 1. We 0
shall associate with the function f a function S{(f) defined on [0,1]' by the formula
Si(f)It = f(tt), where t = {tn}. Observe that the norm of SS(f) in the space N(L(Q,E,/t)) is equal to 1
i = 1, 2, ...
IIS{(f)II = f N(f(t))dt,
(5.6.11)
0
Of course SS(f) can be treated as an independent random variable. Thus, using the classical formula n
E2
n
(Xi-E(X{)) _
E2(X{- E(Xi)) n
we find that, for at >, 0 such that Y at = 1, i=1
n
n
f [ Y at(Si(f)-1)12d/t = Y f
a
t=1 n
i=1
n
aYat f (SI(f)-1)2dp i=1
n
= a f (f(t)-1)2dt,
(5.6.12)
0
where
a = max {a1, ..., an} . By the Schwartz inequality we have n
n 2
f I atS(ft)-1 d/t < f S'at(Si(f)-1)2d/t. fd
i=1
i=1
(5.6.13)
Weak Topologies in Banach Spaces
245
The function N(u) is concave, hence the following inequality results directly from the definition (compare the Jensen inequality for convex functions) n
n
N(' aiui) > i=1
aiN(ui).
(5.6.14)
i=1
As an intermediate consequence of formula (5.6.14), we infer that for each ge N(L(SQ,2,u))r)L(SQ,E,p), we have
IIg! __
6,
(6.4.2)
i.e. there is auk, 0 < /Uk < p such that C (X/Xnk, e, .k) > 6 .
(6.4.3)
This implies that in the quotient space
KaCpkKeCpKK
(6.4.4)
Chapter 6
262
Thus Kb c pK, .
(6.4.5)
Let x be an arbitrary element of a norm less than 6, IIxII < 6. Let Z denote the coset containing x. Since (6.4.5), IIZ/Full < e. Let x0 e Z be such an element that Ilxo/pII < e. This implies that xoepKE.
(6.4.6)
Let us write
x = xo+(x-x0).
(6.4.7)
Then (x-xo) e Xnk and x-xo e KK+4uKE c p(KE+KE)
(6.4.8)
Since the space Xnk is finite-dimensional, by (6.4.1) the set Xnk n (KE+KE) is bounded. Thus it is totally bounded with respect to the set UK,. There-
fore, by Proposition 6.2.3 and by (6.4.8), there is a finite system of points yl, ..., Yn such that m
Ka e
ti=1
(yi+p(Ke+Ke'+Ke)).
Hence the set Kb is totally bounded with respect to the set K = KE+K,' +K,. Since the set Ke constitutes a basis of neighbourhoods of zero, the space X is a Schwartz space. COROLLARY 6.4.3 (Rolewicz, 1961). Let X be an F-space with a basis {en}. Suppose that there is a positive ro such that sup IItxII>eo o
for every x e X, x 0. Then the space X is a Schwartz space if and only if the functions c(X,,,e,t), where X. are the spaces generated by the elements en+1, en+2, ..., are not equicontinuous at 0 for any E.
Basing ourselves on Corollary 6.4.3, we shall give an example of a Schwartz space which is not locally pseudoconvex.
Montel and Schwartz Spaces
263
Example 6.4.4 Let {p,,} be a sequence of real numbers 0 < pn < 1. By 1(p") we denote the space of all sequences x = {xn} such that Ilxll =
Ixnlp" < +oC n=1
with the F-norm Ilxll It is easy to verify that 10'") is an F-space. The sequence {en}, en = {0, ...,0,2,0,. ..}, constitutes a basis in the space n-th place
l(p"). By simple calculation we find that C(Xn, E, t) = 8jtI2n,
where X, is the space spanned by the elements en+I,en+2, ... and
Pn = inf{pt: i > n}. Therefore, by Corollary 6.4.3, the space l('") is a Schwartz space if and only if pn->0. Let us remark that if pk->0, then the space l(p") is not locally pseudoconvex. Indeed, let 6 be an arbitrary positive number. Let xn = {0, ..., 0,
(6)1jp", 0, ...}. Then Ilxnll = 6. Let p be an arbitrary positive number n-th place
and let np be such an index that, for n > np, pn < p12. Let us take the p-convex combination of the elements xnn+1, ..., x.+k. Then n,+k
xn,+i+ - - - +xn,+k
8
()PY.
n,+l
II
k (.4 p
OP )P12
= 6k 1'2 -> oo.
The arbitrariness of p and 6 implies that the space 1(") is not locally pseudoconvex.
6.5. APPROXIMATIVE DIMENSION
Let X be an F*-space. Let A and B be two subsets of the space X. Let
B be a starlike set. Let M(A,B,r) = sup{n: there exist n elements xl, ..., xn E A such that xt-xk 0 EB for f lr k}.
Chapter 6
264
The quantity M(A, B, e) is called the s-capacity of the set A with respect to the set B. Obviously, M(A, B, e) is a non-increasing function of E.
Let
M (A, B) =
ke):
cp (e) - real positive function,
lim M(A(EB,
e)
+ool.
Let us denote by 0 the family of all open sets and by J the family of all compact sets. The family of real functions
M(x)= n n M(A, B) AE6 BECJ
is called the approximative dimension of the space X (see Kolmogorov, 1958).
PROPOSITION 6.5.1. If two F*-spaces X and Y are isomorphic, then
M(X) = M(Y). Proof. Let T be an isomorphism mapping X onto Y. Then a set A is open (compact) if and only if the set T(A) is open (compact). Hence
M (X) = n n M (A, B) AeCj BE6 AdX BEX
= n n M(T(A), T(B)) = M(Y). Ae7 BE6 AcX BcX
PROPOSITION 6.5.2. Let X be a subspace of an F*-space Y. Then
M(X) D M(Y). Proof. Let U be an open set in X. Let x e U and let rx = inf {I Ix-YII: Y 0 U, y e X} .
Let us put
V=!U {zE Y: IIz-xHI < zrx}.
Montel and Schwartz Spaces
265
The set Vc Y is open as a union of open sets. On the other hand, it is easy to verify that U = Vr) X. Then
M(Y)= n n M(A, B) C n n M(AnX, B) AEg BEG
ACYBcY
AE
BEG
AcYBcY
n n M(AnX,BnX)
AEI BEY
AcYBcY
n n M(A, B) = M(X).
Acg BEG
AcX BcX COROLLARY 6.5.3. If
dimzX__

2r1 = inf a° > 0. ZEE
Indeed, let us suppose that there is a sequence {Zn}, Z. a K, such that aZ, ->O. Since the set Kis compact we can assume without loss of generality that {Zn} tends to Z0 e K. Therefore, for sufficiently large n,
IjZn-Z0H < aaz This implies that a°Zn > s aZo, and we obtain a contradiction, because a4,,-->0.
Let us take a finite r1-net in K, Zi, ..., Z' . The definition of r1 implies that there are points xi e Z;, i = 1, 2, ..., n1, such that n
Al = U {x:
Ilx-x;II __
i.e.,
sup
0
N(u) < E
(6.6.16)
p(A)
k
By (6.6.16), for each linear combination f =
ci f , i=1
PN(f)<S <2e. The hypothesis of Lemma 6.6.17 and the above formula imply the theorem.
Turpin (1973)'showed in fact a stronger result, namely that Theorem 6.6.18 is valid for some generalizations of spaces N(L(S2, E, p)). Suppose we are given a space M(ana,n), where m are positive integers and n are non-negative integers (see Example 1.3.9). We say that the space
M(am,n) is regular if, for m < m', the sequence am,n/am,,n is non-increasing. PROPOSITION 6.6.19. If a space M(am,n) is regular, then S(M(am,n ))
_ {{tn}: lim to n-oo
aq,n ap,n
= 0 for some p and all q}. I
Proof. Let Up = {x: IIxjj < 1}. Since the space M(am,n) is regular, we have, for q > p, Sn_1(Uq, Up) = ap,n/aq,n and Proposition 6.6.5 trivially implies the proposition.
Chapter 6
284
COROLLARY 6.6.20. Let {a,}, {bn} be two sequences of reals tending to in-
finity. Then b (M(an )) = M(an) b (M(bn "na)) = M+(bn h/m) = {{tn}: tnbn 1I' --> 0 for some m}. COROLLARY 6.6.21 (Bessaga, Pelczynski and Rolewicz, 1961). There is an
infinite-dimensional Bo space X which is not isomorphic to its product by the one-dimensional space. Proof. Let X = M(expm22°+'). The sequence belongs to p b(M(expm221)), but it does not belong to {exp(-22°+111)}
b(M(expm22"+')).
Now we shall introduce a class of sequences in a certain sense dual to the class b(X). We denote by 6'(A, B) the set of all sequences {tn} such that lim to bn(A, B) = 0. The class b'(A, B) has the following properties :
if A' C A, B' D B, then b'(A', B') j b'(A, B),
(6.6.1') b' (aA, bB) = 6'(A, B) for all scalars a, b different from 0. (6.6.2')
Let
b'(X)=KE9 n nUEQb'(K, u). By a similar argument to that used for b(X) we find that b'(X) is an invariant of linear codimension, which means that if
codim1X__

*
and Ye d1. If Y is a Montel space, then each continuous linear operator trapping X into Y is compact. Proof. By definition there are absolute bases {en} in X and {fn} in Ysuch that (6.7.11) and (6.7.10) hold. Since the bases are absolute, we may assume without loss of generality that the topology in X (in Y) is given by a sequence of pseudonorms {I I
I Im} (resp. {I
I m})such that for x = Jxn en e X(resp. y =
00
yn fn e Y)
n=1
n=1 00
m= 1,2,...,
IIxlIm = f Ixnl Ilenllm, n=1 (resp. oo
IIyAIM =
Iynl Ilenlim,
m= 1,2,...)
n=1
Let T be a continuous linear operator mapping X into Y. Let hn 00
Co
= T(en) = I ti,n f . Of course for each x = j=1
00
oD
T(x)
xn e,, n=1
YxnT(en) =f xn n=1
00
Chapter 6
294
The continuity of the operator T implies that for each p there is a q = q(p) such that cc
C(P)
=
keaq)Ilp IITII(eekll
sup
T Iti,kllllillp
= sup IIekIIq
< +oo
(6.7.12)
We have assumed that Y is a Montel space. Thus to prove the theorem it is enough to show that the operator T maps some neighbourhood of zero UQ. = {x: IIxIlq. < 1 } into a bounded set.
Since Y e d1, there is a p, such that for each p there are io(p) and p2(p) such that
for i > io(p) (6.7.13) On the other hand, X e d2. Take q = q(p1). Thus there is a qo such that (IIf ll;)2 < Ill{II , Illill ,,
for each q2 there is a ko(g2) such that IIekIIq, >
IIekIIq,
for k > k(q2).
IIekIIq,
(6.7.14)
Take q2 = q(p2). Of course, k(q2) depends implicitly on p. By (6.7.13) and (6.7.14) there is a constant L(p) such that Illillp
(
*

*
IIXIIm =
.
(n-11mlxnJ)P,
n=1
with the topology defined by the sequence of p-homogeneous F-pseudonorms {1I 11m} (see Example 1.3.9). It is easy to verify that LP(n-11m) are Schwartz spaces, and thus Montel spaces. This means that every closed
F-Norms and Isometries in F-Spaces
397
bounded set is compact. Moreover, the space LP(n-1l'") has a total family of linear continuous functionals. Thus, by Remark 5.5.6, every compact
set has an extreme point. Hence the spaces LP(n-1/m) have the strong Krein-Milman property. The spaces LP(n-1/m) are not locally convex. Indeed, let en = {0, ... 0, 1, 0, ...}. The sequence {en} tends to 0, because n-th place
IIeni I,n =
(n-IIm)P ->0
form= 1,2,... On the other hand, n
e1+ ... +en n
m
P
i=1
n
P -n-1/m) = n1-P-P/m ->. 00, 1
n
provided p(1+l/m) < 1. Therefore the sequence I
e1+ ... +en } is not
bounded. This implies that the spaces LP(n-11n') are not locally convex. The important class of spaces, namely LP[O, 1], 0 < p < 1, do not have the strong Krein-Milman property. For this reason we shall prove THEOREM 9.3.12 (Rolewicz, 1968). Let (X, II IIx) and (Y, II IIy) be two real locally bounded spaces. Suppose that the norms II IIx and II IIY are concave, i.e., for all x e X, y e Y, the functions IItxl Ix and IIty I IY are concave for pos-
itive t. Then every rotation mapping X onto Y is a linear operator.
Proof. Let r be a positive number such that the set K2, = {x e X: IIxii 2r} is bounded. Such an r obviously exists, since the space X is locally bounded. Using the concavity of the norm, we shall show that sup IIxiix < r.
(9.3.11)
112zIIx<_ r
Suppose that (9.3.11) does not hold. Then there is a sequence {xn} of elements of X such that IIxniix = r and an =
r
21r-I- Xn 2
.I
II
2
I
x
-+r. Therefore
Chapter 9
398
The concavity of the function ItxIIx implies that I Ian xnIJx < 2r. This leads to a contradiction, because the set K2,. is bounded.
If the set K2r is bounded, then the set K2, is also bounded for all s, xIIx. The function n (r) is continuous and
0 < s < r. Let n (r) = sup
I I2zI Ix?
it is strictly increasing, provided r < ro, where K2,, is bounded as follows from (9.3.11). Let us define by induction
n= 1,2,...
rn=n(rn-1)
Obviously, by (9.3.1), ro > r1 > r2 > ... > rn > ... We shall show that lim rn = 0.
(9.3.12)
indeed, suppose that (9.3.12) does not hold, i.e. that
r'= lim rn>0.
(9.3.13)
Since n(r) is strictly increasing, n(r') < r'. The continuity of the function n (r) implies that there is an r > r' such that n (r) < r'. By the definition of r' there is a positive integer n such that rn < r. Hence n(rn) < n(r) < r'. This leads to a contradiction, because n(rn) = rn+l > r'. Let x' and y' be two arbitrary elements of X such that IIx'-y'. < r0/2. Let Ho = fx e X : IIxx'IIx
and IIx-y'Ix *

*
IXI = f Ix(t)IPdt 0
is almost transitive.
The proof follows the same lines as the proof of Theorem 9.6.3; only in formula (9.6.6) it is necessary to replace IITf(x)IIP by IITf(x)II and IIxiiP by IIxII.
F-Norms and Isometries in F-Spaces
413
COROLLARY 9.6.5 (Pelczynski and Rolewicz, 1962). In the spaces LP[O,1],
0 < p < +oo, the standard norms are maximal. PROPOSITION 9.6.6 (Pelczynski and Rolewicz, 1962). There is a normed (non-complete) separable space with a transitive norm which is not a preHilbert space.
Proof. Let LP, 1 *

*
maps the sets Ea = {t: jx(t)j > a} onto the intervals [0, JEaJ) Let
Uxf =f(hx(t)) Obviously, Ux is an isometry and maps the function x(t) on a function x'(t) such that x'(t) is a non-increasing function and
>0 lx '(01
{
=0
for 0 < t < ax, for ax*

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Zelazko, W.: (1960), 'On the Locally Bounded and m-Convex Topological Algebras', Studia Math. 19, 333-356. Zelazko, W.: (1965), 'Metric Generalizations of Banach Algebras', Diss. Math. 47, PWN, Warszawa. Zelazko, W.: (1972), 'A Power Series with a Finite Domain of Convergence', Comm. Math. 15, 115-117. Added in proof:
Dragilev, M.M.: (1983), Bases in Kothe spaces (in Russian), Rostov University Publications.
Subject Index
abscissa of convergence of Dirichlet series 373 absolute
- basis 292 - p-convex hull 298
basic sequence 67
- - represented by a matrix 333 - sequences equivalent 70 basis 67
- absolute 292, 329
absolutely
- block 72
- convergent series 315 - p-convex set 94 - summing operator 317 absorbing set 40 additive operator 36
- functionals 69
admissible pseudonorm 326 affine group property 392 algebra 173, 174
- Schauder 67
- commutative 174
- F* 174 - locally bounded 174 - semisimple 177 almost transitive norm 410 analytic function 124, 125 approximative dimension 264, 268
- - diametral 274 - needle point 241
balanced set 1 Banach space 96
- Hamel 76 - regular 335
- of the type dl 292 ,
----d2 293 - standard 74
- unconditional 329 ,8F* - space 52 block basis 72 Bochner-Lebesgue integral 123 Bo-space 93 Bo -space 93
bonuded
- approximation property 332 - measure 130 - multiplier convergent series 154
- operator 37 - sequence 37
- set 37
bases - equivalent 70
- quasi-equivalent 335 - semi-equivalent 333
Cauchy condition 18 Cauchy-Hadamard formula 376
Subject Index
450
Cauchy sequence 18 C-bounding point 222 centre of a Hilbert scale 340
-----finite 340
-----infinite 140 chain 46 characteristic function of a random variable 146 C-internal point 222 closed convex hull 223 cluster point
--of aset 33
- - of a family of sets 33 compact -measure 132
- operator 206, 288 complementary function to a convex
- transitive norm 415 co-universal space 63 C-sequence 165 C*-space 73 C-space 166
8-divergent zone 245
derivative of a vector valued function 198 diametral approximative dimension 274 Dirichlet series 371
- -entire 371 distribution of a random variable 146 domain
- of an operator 35 - proper of an integral operator 84 dominated convergence theorem 84
function 199 complete
- space 18 - set in a linear topological space 34 completion
- of a linear topological space 35 - of a metric space 21 complex rational numbers 28 condition
- Cauchy 18 - (JE) 10
-(A,) 11 -(1Q) 12 -(0) 173 - R 288 conjugate space 39, 199 consistent family of F-norms 46 continuous linear
- - functional 39 - - operator 39 continuously imbedded subspace 77 convergent sequence 2 convex
- function 111 - hull 221 - set 89, 221
element invertible 175 e-capacity 264 e-net 265 equicontinuous family of operators 39 equivalent - bases 70
- basic sequence 70
- F-norms 5 - metrics 2 extreme - point 238 - subset 238
finite
- centre of a Hilbert scale 340 - dimensional operator 206
- e-net 265 finitely supported function 46 F*-algebra 174 O-operator 294
F-norm 4 F-norms equivalent 5 F-pseudonorm 15
Subject Index F-space 22 F*-space 5 F*-space quotient 5 functional
451
identically distributed random variables 183 independent random
- - measures 145
- - variables 138
- continuous linear 39 - dimension 369
index 290
- linear 39
inequality
- Minkowski 188 - multiplicative linear 176 - non-trivial linear 75, 187 function
- analytic 124, 125 - complementary to a convex function 199 - convex 111
- finitely supported 46 - measurable 133 - Riemann integrable 120 functions
- equivalent on an interval (0, +oo) 112
- equivalent at infinity 112
--at0 112 fundamental
- family of sets 33 - sequence 18
F-closed set 226 F-closure 226 -compact set 226 F-continuous functional 226 F-topology 226 G. -set 20
Haar system 75 Hamel basis 76 Hermitian derivative 350 Hilbert scale 340 Hilbert-Schmidt operator 320 Hilbert space 22 homogeneous random measure 146
- Kolmogorov-Kchintchin 169
- Markov 346 - Paley-Zygmund 138 - Tchebyscheff 137
-Young 199 integral
- Bochner-Lebesque 123 - of a simple function with respect to a vector measure 134
- Riemann 120, 126 integration of a scalar valued function with respect to L'-bounded measure 143, 145
invariant metric 2 inverse 175 invertible element 175 isometry 390 isomorphic spaces 44
Kolmogorov-Kchintchin inequality 169 Kothe power spaces 285
- - - of finite type 285 - - - of infinite type 285 Levy-Kchintchin formula 147 linear
- codimension 62 - dimension 45 - functional 39
- operator 36 - space 1 - topological space 33, 223 locally
Subject Index
452
- bounded algebra 174
- - space 95
- compact space 250 - convex space 93, 225 - p-convex space 90 - pseudoconvex space 90, 93
multiplication 173 mulitplicative-linear functional 176
near isomorphism 290 needle point 241
- - approximative 241
- - space 250 Markov inequality 346 maximal norm 409 M-basis sequence 76 measurable function 133 measure 128 - bounded 130
-compact 132 - independent random 145 - Lm-bounded 135 - non-atomic vector 145
- separable 27 - or-finite 10
- space 10 - variation of 128 - vector valued 128 metric 1
- invariant 2 - linear space 1 - stronger than 2 metric equivalent 2 metrizable topological linear space 33 metrized modular space 10 metrizing modular 6 Minkowski functional 188 m-quasi-basis 76 modular 6
- metrizing 6 - space 10 modulus of concavity
- - - of a set 89 - - - of a space 96 monotone convergence theorem 83
- norm 48
Montel space 251
non-atomic vector measure 145 non-decreasing norm 7 non-trivial continuous linear functional 75, 187
norm 4 - almost transitive 410 - convex transitive 415
- equivalent to 5 - maximal 409 - monotone 48
- non-decreasing 7 - nuclear 309 - of a basis 103 - submultiplicative 174 - symmetric 421, 425 - stronger than 5 - transitive 410 normal sequence of subdivisions 120 normed space 96 nowhere dense set 18 nuclear
- norm 309 - operator 308
- space 296
operator - absolutely summing 317
- additive 36 - bounded 37
- compact 206, 288 - continuous linear 36 - finite dimensional 206 - Hilbert-Schmidt 320
- linear 36
Subject Index
- nuclear 308 Orlicz space 11
Paley-Zygmunt inequality 138 perfectly bounded sequence 173 p-homogeneous pseudonorm 90 plane of symmetry 420 point extremal 238 polynomials Tschebyscheff 137 precompact set 255 pre-Hilbert space 18 product space 6 proper domain of an integral operator 84 property P 135
453
Schwartz space 256 section 46 semi-equivalent bases 335 semisimple algebra 177 separable
- measure 27 - space 26 sequence
- bounded 37 - Cauchy 18 - convergent 2 - fundamental 18 - linearly independent 75 - linearly m-independent 76
- of independent random variables 138, 183
pseudoconvex set 89
- perfectly bounded 173 - Rademacher 138 - topologically linearly independent
pseudonorm 93
- admissible 326 - p-homogeneous 90
76
- weakly convergent 76, 130 quasi-norm 211, 252 quasi-equivalent bases 335 quotient
series
- absolutely convergent 315
- bounded multiplier convergent 154 - unconditionally convergent 152
- F*-space 5
- space 5
set
- absolutely p-convex 94
- absorbing 40 Rademacher sequence 138 random variable 136, 183
- - symmetric 147, 183
- balanced I
- bounded 37 - convex 89
- - identically distributed 183
- F-closed 226 - F-compact 226
- - independent 138, 183
- Gd 20
random variables
reflexive space 229
regular
- basis 333 - space 283
Riemann
- integrable function 120 - integral 120, 126 rigid space 210 rotation 390
- nowhere dense 18 - of the first category 18 - of the second category 18 - precompact 255 - pseudoconvex 89
- or-finite 10
- solid 77 - starlike 89 - starshaped 89
Subject Index
454
- totally bounded 255 - tree-like 46 - unfriendly 79 smooth point 401 solid
- set 77 - space 77
- LV(Q, --,.u) 12
- L°(n, E, u) 16,
-L(S1,I, u) 14
- metric 1 - metric linear 1 - metrizable 33 - metrized modular 10
- Banach 96
- modular 10 - Montel 251
- BB 93 - B° 93
- M 14
-,6F* 52
- M [a, b] 14 - M (am,n) 17
space
- complete metric 18
-m 14
-conjugate 39, 199
- M (S1, E, u) 14 - needle point 242
- c 15 - c° 15 - C(SI) 14
-N(l) 13
space complete metric linear 19
- C (S1/S1°) 15
- C [a, b] 15
- normed 96 - nuclear 296
- N (L) 13 - N (L[a, b]) 13
- C°°(S1) 16
- N(L (.Q, .E, µ)) 11
- e,(D) 16
- of Dirichlet series 371
- couniversal 63
- of the type dl 292 - of the type d8 293 - of the type d;, i = 1, 2 338 - Orlicz 11 - pre-Hilbert 18
-F 22
- F* 5
- having trivial dual 39
- Hilbert 22 - -)f (D) 354 - =l((D) 355 - C1 355 - Kothe power of finite type 285 - Kothe power of infinite type 285
- quotient 5 - reflexive 229
- regular 283
-rigid 210
- linear 1 - - topological 33, 223
- Schwartz 256 - second conjugate 229 - separable 26
- locally bounded 95
- solid 77
- - compact 250 - - convex 93, 225 - - p-convex 90
- strictly galbed 157
- - pseudoconvex 90, 93
- S(S1, E, µ) 12 - c5(En) 17
1P 13
- LP 12
-(s) 12 - S [a, b] 12
- LP [a, b] 12
- S(,) 371 - S(A) (R) 373
- LP am,n) 17
- S('A.) 375
Subject Index
- S""1 375
- a 340 - topological linear 33, 223
- universal 45 - - with respect to isometry 436
- - - - - isomorphism 45 - - - - - linear codimension 63
455
topology of bounded convergence 39 totally bounded set 255 total family of linear functionals 44, 195, 226
transitive norm 410 tree-like set 46 triangle inequality 1
- - - - - - dimension 45
- with arbitrarily short lines 196
- - bounded norms 389 - - non-trivial dual 187 - without arbitrarily short lines 52 - with strong Krein-Milman property 391
- with trivial dual 187 - (Xi)(.) 31 spaces
- isomorphic 44 - nearly isomorphic 290 sprctrum 181 standard basis 74 starlike set 89 starshaped set 89 subgroup
unconditional basis 329 unconditionally convergent series 152 unfriendly set 79 unit of na algebra 174 universal space 45
- - with respect to isometry 426
- - - -- - isomorhism 45 - - - - - linear codimension 63 - - - - - linavr dimension 45 variation of a vector valued measure 128 vector valued measure 128
- equicontinuous 391
- fat 391 submultiplicative norm 174 subspace 5 surjection 400 symmetric norm 421, 425 Tchebyscheff inequality 137
weakly convergent sequence 76, 230 weak - topology 226
- - of functionals 226 - *-topology 226
- polynomials 137 topological linear space 33, 223
Young inequality 199
Author Index
Alaoglu, L. 228, 434 Albinus, G. 95, 434 Antosik, P. 350, 434 Aoki, T. 95, 434 Arnold, L. 124, 434
Douady, A. 291, 436 Dragilev, M.M. 286, 287, 335--338, 436,
Aronszajn, N. 79, 80, 85-87, 434 Atkinson, F.V. 291, 434 Auerbach, H. 298, 408, 434
Dunford, N.S. 202, 445, 435, 437 Dubinsky, E. 238, 307, 331, 437 Duren, P.L. 193, 437 Dvoretzky, A. 316, 235, 437 Dynin, A. 313, 328, 330, 437
448
Drewnowski, L. 76, 77, 124, 133, 436, 437
Banach, S. 5, 39, 42, 43, 86, 100, 101, 168, 410, 427, 434, 435
Baire, R. 18 Bartle, R. 435 Beck, A. 435 Bessaga, C. 52, 65, 66, 107, 167, 168, 196, 237, 238, 252, 254, 274, 275, 277, 385, 389, 427, 428, 435
Bohnenblust, H.F. 191, 445 Bourgin, D.G. 252, 435 Burzyk, J. 307, 436
Charzyfiski, Z. 390, 436 Cowie, E.R. 416, 436 Crone, E.R. 335, 436
Day, M.M. 195, 436 Dieudonne, J. 252, 254, 436 Djakov, P.V. 331, 333, 436
Eberlain, W.F. 231, 437 Eidelheit, M. 205, 250, 385, 437 Egorov, D.F. 134
Fenske, Ch. 305, 437 Figiel, T. 238, 401, 438 Frechet, M. II, 438
Gawurin, M.K. 438 Gelfand, I.M. 176, 369, 438 Gohberg, I.C. 290, 438 Goldberg, A.A. 379, 382, 438 Goldstine, H.H. 229, 438 Gramsch, B. 124, 183, 438 Grothendieck, A. 317, 438 Gri nbaum, B. 438 Gurarij, W.I. 412, 439
Author Index Hahn, H. 189, 439 Henkin, M.G. 296, 365, 442, 443 Holsztyfiski, W. 439 Hyers, D.M. 211, 252, 439
Iyachen, S.O. 439
James, R.C. 234, 235, 439
Kakutani, S. 2, 439
Kalton, N.J. 46, 47, 51-53, 63, 76, 99, 193, 197, 199, 206, 210, 211, 213, 220, 248, 307, 419, 420, 439, 440 Klee, V. 19, 76, 197, 440
Klein, Ch. 120, 440 Kolmogorov, A.N. 96, 264, 440 Komura, T. 341, 440 Komura, Y. 341, 370, 371, 440 Kondakov, V.P. 333, 335, 336, 440 Krasnosielski, M.A. 202, 440 Krein, M.G. 103, 240, 290, 440 Krein, S.G. 340, 438, 440 Kwapiefi, S. 169, 440
457
Mazur, S. 21, 39, 40, 100, 101, 116--118, 121, 161, 168, 176, 202, 250, 251, 385, 390, 401, 427,442, 434, 435, 437 Metzler, R.C. 162, 442 Mikusifiski, J. 350, 434 Milman, D.P. 103, 240, 440 Mityagin, B.S. 274, 275, 277, 296, 313, 314, 328, 330, 331, 340, 354, 365, 436, 442, 443
Moscatelli, B. 307, 326, 443 Musial, K. 140, 443 Musielak, J. 6, 8, 443
Nakano, H. 6, 443 Ogrodzka, Z. 349, 443 Orlicz, W. 6, 8, 39, 40, 101, 116-118, 121, 153, 161, 169, 173, 202, 442, 443
Paley, R. 138, 443 Pallaschke, D. 206, 208, 443 Peck, N.T. 76, 197, 211, 213, 248, 439, 443
Pelczyfiski, A. 46, 52, 106, 107, 167, 168, 196, 237, 238, 268, 274, 275, 277, 321,
Labuda, I. 76, 437, 441 Landsberg, M. 94, 441 Lebesgue, H. 134 Leray, J. 224, 441 Levi, E.E. 182 Ligaud, J.P. 297, 298, 305, 441 Lindenstrauss, J. 321, 441 Lipecki, Z. 76, 441 Lorch, E.R. 441 Lusky, W. 412; 441 LuxemburgW.A.J. 77,79,82--84,441 Mankiewicz, P. 391--393, 441 Marcus, M. 186, 441 Matuszewska, W. 136, 442
330, 385, 389, 410, 412-415, 418-420,435,443,444 Petrov, V.V. 184, 186, 444 Pettis, B.J. 444 Phelps, R.R. 401, 444 Pietsch; A. 321, 324, 444 Pisier, G. 136, 442 Popov, M.M. 63, 444 Prekopa, A. 146, 444 Przeworska-Rolewicz, D. 124, 183, 290, 444
Raikov, D.A. 444 Ramanujan, M.S. 307, 437, 444 Retheford, J.R. 435
Author Index
458
Ritt, J.F. 384, 444 Roberts, J.W. 210, 211, 220, 241, 242,
Szeptycki, P. 77, 79, 80, 85--87, 446 Szlenk, W. 446
245, 246, 248, 439, 444
Robinson, W.B. 307, 335, 436, 437 Rogers, C.A. 316, 325, 437 Rolewicz, S. 14, 52, 89, 95, 96, 106, 108, 113, 120, 124, 135, 155, 183, 193, 194, 196,
198, 238, 252, 254, 260--262,
274, 275, 277, 290, 355, 376, 385, 389,
Talagrand, M. 131 Tichomirov, W.M. 274, 440, 446 Tillman H.G. 446 Turpin, Ph. 64, 88, 126, 127, 131, 135, 157,16392789280-283,446,447
410, 412--415, 418--420, 435, 440, 444,445
Romberg, R.G. 193, 437 Rosenberger, B. 236, 445 Rutitski, Ja.B. 202, 440 Rutman, L.A. 103, 440 Ryll-Nardzewski, C. 135, 140, 141, 155,
Ulam, S. 390, 442 Urbanik, K. 150, 447 Vilenkin, N.Ja. 369, 438 Vogt, D. 332, 447
186, 254, 443, 445
Schauder J. 67, 445 Schields, A.L. 193, 437 Schock, E. 305, 437 Schwartz, J. 202, 435, 437 Schwartz, L. 169, 171-173, 445 Semadeni, Z. 238, 444 Shapiro, J.H. 193, 197, 206, 248, 439, 445 Sierpinski, W. 20, 446 Simmons, S. 28, 446 Singer, I. 330, 421, 444, 446 Slowikowski, W. 258, 446 9mulian, V. 231, 446 Sobczyk, A. 191, 435 Srinivasan, V.K. 376 Steinhaus, H. 39, 435 Sternbach, L. 21, 442 Stiles, W.J. 101, 103, 446 Sundaresan, K. 184
Waelbroeck, L. 126, 127, 210, 447 Whitley, R.J. 231, 447 Wiener, N. 181 Wiliamson, J.H. 291, 447 Wobst R., 390, 447 Wojtyriski, W. 330, 340, 447
Wood, G.V. 417-421, 447 Woyczynski, W. 140, 141, 150, 152, 184, 186,440,441,443,445--448
Zaanen, A.C. 82--84, 441 Zahariuta, V.P. 288, 291, 295, 338, 340, 365, 368, 448
Zobin, N.M. 331, 443 Zygmund, A. 138, 443 2e1azko, W. 90, 124, 174, 176, 177, 181, 182, 448
List of Symbols
A+B 1 p (x, y) I
M(E) 128 E(X) 137
(m1),...,(m3) 1 IA I
V(X) 137 A(X) 157
(X, p) 1
conv(A) 221
x.--->x 2
conv(A) 223
P
xn --> x 2
n(X) 229
Ilxll 4 (n 1), ... , (n 6) 4 (X II II)5 X/Y 5 (md 1), ... , (md 5) 6
X* * 229 E(K) 238
(i 1), ... , (i 5) 17
M(A, B, e) 263 M(A, B) 264 M(X) 264 M'(X) 268 M;(X) 268 M(X) 270 6 (A, B, L) 274
G8 20
6n(A, B) 274
(X3(,) 31, 100
6(X) 274 a(X) 275 6'(A, B) 284 6'(X) 284 6'(X) 284 T(X) 295
X, 6
(md 5') 6 (x, y) 17
DA 36
Y) 38 Be(X B0(X) 38
B,(X - Y) 39 Y) 39 X* 38, 199 dimiX 45 E [c] 46 B (X -
dn(T) 308
a(T) 318
suppx 47
Ct,t 327
codimiX 62
DK 84 c(A) 89 c(X) 96
d; 338 H(X) 391 T(X) 391 Lin (X) 392 Aff(X) 392 Inv(G) 391
n(t) 107
G(II
c , 77 E.'\, o 81 X, 81
ID 408

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